Your scepticism is well considered as both terms carry different meanings amongst historians depending on the context in which it is used. Defining a "unfree peasant" and a "free peasant" can be a challenging task for medievalists - there's a lot of nuance to consider. I'll start by defining an unfree peasant, delve into the complexities of that classification, and then use that to conclude with how these complexities translate to the consideration as to what a free peasant is.
But before I continue I need to say that I am not well versed in the situation of the 'peasant classes' outside of Medieval Europe - so I won’t speak beyond that area. But I would like to briefly acknowledge (and encourage others to elaborate on) the disparate and varied conditions of the tenant labourer & service classes outside of Europe - as they were myriad and should be brought into this discussion by those more familiar with the subject than I.
An unfree peasant is generally considered by medievalists to be any peasant that was, well, not free. This could mean several things; the most common form of an unfree peasant would fall under the catchall term 'serf'. Trying to define a serf universally is essentially impossible; as a serf in Russia in the 1600s lived under very different conditions and circumstances than a serf in the Holy Roman Empire during the 10th century. But while we can't define them universally we can try to define them generally: A serf is a peasant farmer who was bound to the land they lived on, usually from birth, and could be brought or sold along with that land (exceptions to be noted later), had little to no rights over their bodies, and could not leave the land they were bound to or marry without their lord's permission. This condition (at least in Western Europe) is commonly believed to have evolved from the Colonus/Coloni system used in the late Roman Empire. But it is important to note that while the condition of serfdom was similar to slavery - it categorically was not. Slaves were a distinct social class within society, and had even less rights, privileges, and protections than the average serf.
But while this is the 'golden standard' of an unfree peasant, let us define some of the different categories and contexts in which the term was used. This will, obviously, not be a comprehensive list.
As we can see, the distinction between the two groups – free and unfree – can be foggy. Is a peasant free just because they are acknowledged as such outside the authority of their lord? Or do they require the right to leave their parcel of land freely? What if they can leave their land but lack any rights or freedoms outside of their lord’s protection? And then at what point does an unfree peasant just become a slave – what is the threshold at which indentured servitude becomes something far worse? Medievalists have spent a lot of time making these distinctions and trying to settle on clear categorical definitions – but the countless styles of peasantry and serfdom make it quite the challenge. Matters only become worse when we consider broader historical contexts.
For example; many historians would define an individual living in, say, 17th century England that had the most enviable of legal rights and conditions by the standards of their 11th century counterpart as still being unfree, for when compared to their contemporaries they likely would have been perceived as such. Now that isn’t to say that a tenant farmer in the 17th century was living a life resplendent with freedoms – but we need to remember that what freedom is and isn’t is often defined against, and within the context of, what is considered unfree. Contemporary primary source accounts only further complicate this issue, especially in later periods: what did freedom mean for a revolutionary pamphleteer in 18th century France? What would the positions of a probably-privileged-writer in England be when using terms like “free” and “unfree” to define the peasant classes when writing about, say, the Enclosure Acts – which removed free access to the common lands from many farmers, but didn’t remove, say, their bodily autonomy, or right to marry without a lord’s permission?
But then, to conclude – what is a “free peasant”. Now, the common responses to this question would be; Boyers, Freeholders, Yeomen and the various other free-tenant classes that generally weren’t to be traded or purchased, bound to land, nor sometimes even lord. But I hope the earlier parts of this answer lead you to consider this answer with scepticism. Because while these individuals had more rights than their peers, there are medievalists that would still complicate this matter further – in particular, due to the lowly ‘vagabond’. For while the vagabond lacked legal protections, was often persecuted, and tended to a hard life – they possessed a remarkable amount of independence and freedom from lords, laws, and implicit bondage:
At the core of the concept of vagabondage were two key elements—voluntary unemployment and itinerancy—both of which, historically, were connected with labor and residence obligations of medieval serfdom… after 1500, western European governments continued to enforce policies against vagabondage, with few other means by which to control and harvest the labor of an otherwise independent and disordered underclass.
A. L. Beir. Masterless Men: The Vagrancy Problem in England, 1560–1640, 1985.
So who is more free? Someone who is safe and has protections under the law, but carries with those rights various restrictions and bondages to lord and rent – if not land? Or someone who holds almost no protections or safety under the law, but may live with far more independence and separation from lord and lease? I can’t answer this question for you – but know that whatever answer you choose, it is born out of the context you live in and the breadth of your experience – as it would when any medievalist considers which peoples were, or were not, qualified as “free”.