I can't help answer your question, but I would like to add that the image you've linked to is only the top part of a 1647 broadsheet which also included verses about each group. The full text of the broadsheet can be found here.

This is a really interesting question! I am not sure if I will be able to provide information about every image but I can hopefully shed some light on what the purpose of this document was and why it contains such a seemingly baffling array of lesser-known and out of place sects.

Firstly, although its a good guess, this isn't from the Elizabethan or Jacobean period, but from 1647, at the height of the Civil Wars, when both political and religious order had largely broken down and all manner of new and strange dissenting Protestant sects and political movements start to appear in sources. Some of these groups were real and genuinely influential, others were real but their numbers were exaggerated, and others appear to have been almost entirely fictional.

As you alluded to in your question, many of these groups carried names which would more immediately be associated with late antiquity or the early medieval period than the early modern. Partly, this was because many Protestant dissenters had a genuine interest in emulating the early church and returning institutions like the episcopacy to their pure 'apostolic' form. However, as I already mentioned, there was also a great degree of exaggeration and panic regarding these new sects, and writers and pamphleteers often applied lurid names which evoked ancient heresy to groups which did not really have much in common with their namesakes at all.

Because of this, the way in which I would advise approaching this image is not so much as a literal catalogue of sects and opinions in England at the time, but rather to think of each image as a caricature of certain trends and ways of thinking that the author wishes to attack. As another poster pointed out, there are actually poetic verses which go along with each of these images, which help shed further light on their meaning.

So with that out the way, let's see what I can say about the images. I don't sadly have time to go through all of them in detail, but I will try and provide some information about the ones that stand out to me.

Firstly, the Jesuit. The Jesuits, were, and still are, a real Catholic missionary order, long feared and despised in Protestant England as seditious agents of the Pope. Their significance in the 1640s went beyond this, however, and the word became an embodiment more generally of religious tyranny over civil matters. The idea was that what made the Jesuits so dangerous was a belief in the 'deposing power' of the Pope, in other words, the ability to dethrone kings and princes. This concept became extended even to Protestant groups who were seen as using religion to usurp civil authority, and it is not uncommon in this period to find different groups of Protestants accusing each other of behaving in 'Jesuitical' fashion. This is encapsulated in the opening couplet of the accompany verse regarding Jesuits:

By hellish wiles the States to ruine bring,

My Tenents are to murder Prince or King:

The next image does not explicitly have a label on the image beyond a description of them as Welsh and blasphemous, but from the verse appears to be a Socinian. which was a nontrinitarian doctrine which emerged in the 16th century, but which was also became common in England as a catch all term for dissenters viewed with suspicion, and I suspect that is how it is used here, with the relevant verse rather vaguely accusing the Socinian of trickery and 'strange notions':

By cunning art my way's more nearly spun,

Although destructive to profession;

Obscuring truths, although substantiall,

To puzle Christians or to make them fall:

That precious time may not be well improv'd,

Ile multiply strange notions for the lewd.

The next one is the Arminian, which slightly confusingly does not have anything to do with the country of Armenia, but instead derives from the name of the Dutch theologian Jacobus Arminius. Arminianism was a form of Reformed Protestantism which differed from classical Calvinism on issues like predestination, which is the topic of the verse attached to this image:

Would any comfortlesse both live and die?

Let him learne free wills great uncertaintie:

Salvation that doth unmov'd remaine,

Arminian Logick would most maintaine,

And faith that's founded on a firme decree,

Is plac't by them to cause uncertaintie.

The relevance of Arminianism to early modern England is that many saw it as the theological inspiration behind the reforms undertaken by Archbishop William Laud.

Laud was Charles I's Archbishop of Canterbury who oversaw attempts at various reforms to the English and Scottish churches, which were often unpopular and contributed to the outbreak of the Civil Wars. The extent to which Laud's reforms genuinely represented an attempt to impose Arminianism on the British kingdoms is still a matter of debate, but readers in 1647 would likely associate it with the reforms which brought the kingdoms into chaos.

Next is the Arian, which as you rightly pointed out, seems about a thousand years out of place at this point. There were, in fact, some genuine attempts at reviving Arianism in early modern England. The priest John Assheton who recanted in front of Thomas Cranmer in 1548 has been labelled as an Arian, although others have argued his views were not true Arianism but more closely resembled Socinianism, or some other form of antitrinitarianism. In any case, as I mentioned at the start, there was often rhetorical value in associating one's opponents with known forms of heresy.

The Adamites, as you guessed, genuinely did go naked during their services. However, much like the Arians, they were primarily a phenomenon of late antiquity rather than the early modern period. While there were reports of Adamite revivals in the Reformation period elsewhere in Europe, I have not before encountered a reference to them appearing in England, and from a quick search can find no other reference to this besides other discussions of this image. Without further evidence I'd have to assume this is probably more of a rhetorical flourish than a reference to genuine Adamites, though in the chaos of the Civil Wars, I suppose its possible there genuinely were attempts to revive Adamitism.

This answer is getting very long and frankly, I would be drifting into guesswork were I to attempt to go into detail about some of the even more obscure sects that the image and poem mentions (the poem has even more sects not featured in the image), which isn't the point of this sub.

A couple of final points before I stop though. 'Soule Sleepers' refers to 'Christian mortalism', which is the belief that the soul is not conscious prior to resurrection, though I am afraid I know nothing about the role of this belief in early modern England. As for the background of this image, I can find no record of an author, but printed propaganda of this sort was common in the 1640s, and its purpose, ultimately, was likely to advance the religious and political agenda of its author.

Precisely what agenda this was is hard to be certain of without an author, but given that it attacks Jesuits, Arminians, Independents and dissenters, my speculation would be that this image and poem were created by a Presbyterian. The Presbyterians can be seen broadly as the 'moderate' faction within the Parliamentarian alliance, who rebelled against the king initially but by 1647 had become alienated by what they saw as the extremism of the Puritan/Independent faction (the most famous member of which was Cromwell).

If there are any particular parts of the rest of the image or poem that you're interested in though, do please ask and I'll see if I have anything to say on them!

The image is at times satirical, absolutely propaganda. Here is a transcription of the text on the woodprint, as well as the writings within about each of the different groups. What follows is my interpretation in light of those inscriptions and other general knowledge from the period.

All of the sects were outlawed and suppressed at the time, and most likely, this print was part of a chapbook (small, easily printed booklet used for ideological purposes) used to help common people know how to identify these sects and refute their beliefs.

Briefly, I'll give a quick outline of the ones I recognize and update the list when/if I find information on the ones I don't.

1. Jesuit. A Roman Catholic religious order founded by Ignatius of Loyola in the 16th century. While this order was not founded specifically in response to the Protestant Reformation, it took on a role of combatting Protestantism throughout Europe. Most likely, they found their way onto this list because, for a variety of reasons, Jesuits became labeled as spies. One reason was their quick ascendancy within the Roman Catholic church, a meteoric rise that saw them involved in everything, everywhere. Their popular role as confessors (people who heard private confession) to nobles was particularly alarming. Another was that Jesuits had the habit of, well, not wearing a distinctive habit (religious garb, like a monk's cowl), and so could not be readily identified. All of these things birthed the fear of Jesuit conspiracies, especially among Protestants. Such anxieties were heightened in England with the fear of Roman Catholic rule returning as it did under Mary Tudor, and the plots and rebellions against Elizabeth by Mary, Queen of Scots, among others. Point being, a Protestant England was by no means a guarantee, and the Jesuits were feared to be the secret Roman Catholic operatives at work trying to regain England for Rome. There was, of course, plenty of animosity toward the Jesuits among Roman Catholics at this time, but that's a story for another day.
2. Some guy name Evins in Wales who said he was Christ.
3. Arminian. A reference to followers of Jacobus Arminius, who held that people had the free will to chose salvation or damnation by either responding to or resisting the grace of God. This arose as a response to Calvinism, which denied the freedom of the will and held instead a belief in double predestination, meaning that both those who were saved and those who were damned were elected so by God. The inclusion of Arminians on this list makes it quite likely that this was a Calvinist-leaning publication.
4. Arian. You are right that Arianism mostly died out in the early middle ages, but that did not stop people from accusing others of Arianism, similar to how Martin Luther and the other reformers accused the Roman Catholic church of their time of being (semi-)Pelagians. Arianism was a popular charge to use against others, particular in matters of Eucharistic theology, where differences in belief over how Christ was present in the Eucharist led polemicists to accuse their opponents of denying the full divinity of Christ (the cardinal sin of Arianism).
5. Adamite. Another heresy from Late Antiquity, this one referring to groups of Christians who worshipped in the nude. Believing that Christ had returned them to a pre-Fall state (that is, the state of being which Adam and Eve enjoyed before the Fall), they rejected clothing in worship, as Christians generally interpret Adam and Eve's shame at their nakedness in Genesis to be a consequence of their sin. There were reports that such practices had reemerged in the late middle ages, and while this was possible, most likely these anxieties were the product of salacious rumors.
6. Libertin[e]. Christians who held that what it means for Christ to have fulfilled the law is that Christians no longer had a need of it, for they were governed by the Holy Spirit and lived perfectly free. Plenty of theologians taught something like this, in that someone filled with neighborly love did not need a law telling them not to steal or murder, etc. This accusation was used for people who took that thinking to an extreme, and thus was a fairly broad-brush accusation. It could cover anything from anarchism to denying that certain laws of the Old Testament (e.g. prohibitions against images) were binding on the Christians. The image shown is of someone smashing the Ten Commandments, which likely puts it closer to the latter category.
7. Antescripturian. Literally meaning before (or without) Scripture, this is a pejorative label for those who denied that Scripture could be called the Word of God on the basis that Christ alone was the Word of God. Thus, they held that the Bible should be viewed as any other book written by humans and cannot be used as the basis for infallible doctrine. This view should not be confused with Roman Catholicism despite Protestant accusations about their approach to the authority of Scripture.
8. Soule-Sleeper. Technically, this refers to the belief that the soul "falls asleep" upon death, to be awakened at the second coming of Christ for final judgment, only then to experience heaven or hell. This is as opposed to the belief that upon death a person experiences judgment immediately and is sent to heaven or hell, only to have that judgment affirmed at Christ's return. Based on the text inside, however, it seems that here it is associated with the rejection of the immortality of the soul and the afterlife, and thus a certain hedonistic way of living.
9. Anabaptist. Literally meaning "rebaptizer," it is a movement commonly associated with Zürich in Switzerland but which scholars believe had multiple origin points. Anabaptism is defined by the rejection of infant baptism as a valid or true baptism. They practiced what was called "believer's baptism," which meant being baptized at an age when one could profess faith for oneself. This meant practically that its adherants rejected their baptism as infants and were baptized "again" as adults (hence the name, which was pejorative), though of course they believed that their adult baptism was their only true baptism. Easily the most highly persecuted sect in the 16th century, as both Roman Catholics and magisterial reform movements (Luther's, Calvin's, Zwingli's, etc) seemed to compete at who could exile and/or kill the most of them.
10. Familists. Also known as Niclaes after their founder, Henry Nicholis, the Familia Caritatis (literally, Family of Love) was a mystical sect founded in the 16th century. They also rejected infant baptism (like the Anabaptists) and further rejected the doctrine of the Trinity. They also held that God did not rule directly but rather indirectly through the natural order. This text seems particularly focused on their rejection of the Sabbath (meaning, setting aside one day for worship) and their presumption to enjoy a state of pre-fall perfection like the Adamists (but without the nudity, it seems).
11. Seeker. Though unfamiliar with this group, based on the text, it seems to be a reference to those who rejected the orders of the church (priests, bishops, etc.) and the Scriptures in favor of personal revelation and present-day miracles. This would make them similar to the Antescriptursians above and particularly reminiscent of what we find in Germany in the 16th century with the likes of Thomas Müntzer and other so-called Enthusiasts, who subjected the authority of Scripture to the living voice of the Spirit speaking directly to the believer. Given the role of Müntzer in the German Peasants' War, fear of such beliefs spread widely.
12. Divorcer. As the name suggests, those who held that divorce should be permissible outside cases of adultery. Not a popular stance, even after the whole Henry VIII thing.

As the link above shows, there are more listed in the chapbook than what are in the woodcut, but I'll stop there. If you see more sects you want to know about, just comment, and I'll see what I can do.

SELECTED SOURCES

The Radical Reformation by George Hunston Williams

The Swiss Reformation by Bruce Gordon

A Companion to Anabaptism and Spiritualism, 1521-1700 edited by J.M. Stayer.

The English Bible in the Early Modern World edited by Robert Armstrong, Tadhg Ó Hannracháin.

Reformations: The Early Modern World, 1450-1650 by Carlos Eire

*The Collected Works of Thomas Muntzer" edited by Peter Matheson