Stahl's phlogiston came, by his own account, from Johann Becher. Becher was a mid-17th century alchemist and chymist who in many ways lived up to the early modern stereotype of the alchemist-charlatan (as in Ben Johnson's play), for instance, Becher had encountered a rock which made you invisible if you held it, and tried to get the Dutch government to sponsor his scheme to produce gold from sand. (That said, one should avoid adopting the traditional Enlightenment historigraphy of viewing alchemists as little more than charlatans)

Becher's legacy would be his book Physica Subterranea ('Subterranean physics') of 1669, where he laid out his ideas, which were not really that revolutionary. Becher held fast to the classical four elements, but considered 'air' and 'fire' to be the agents of chemical change rather than elements per se. Earth was the most significant element (perhaps inspired by his professional interests in minerals and mining). He subdivided the element of earth into three 'earths': terra lapidea (rocky earth), terra fluida (liquid earth) and terra pinguis (oily earth), which bore more than a passing resemblance to the Three Principles of Paracelsus.

The association between 'oiliness' and fire might not seem so obvious to a modern observer, but at the time it was 'established' that sulphur was the element of fire - it came out of the earth near volcanos, it burned with a mysterious blue flame and left no ash, and such. But sulphur is also a yellow, waxy substance in elemental form, and oil of vitriol (sulphuric acid) is of course an oily substance. (Compare to the 'sulphur principle' of Homberg, which he associated with light)

It doesn't seem like Becher's book had much impact - Boyle never mentioned it. Becher on the other hand, was influenced by Boyle. (In fact, those who would paint Boyle as the 'father of chemistry', particularly 19th century Englishmen, would often paint phlogiston almost as a distraction that set chemistry back between Boyle and Lavoisier) Even though Stahl gave huge credit to Becher, phlogiston theory was largely his own invention. (to some extent he was probaly following the old routine of seeking raising his own status by raising that of his predecessor)

There is no way to explain what exactly whas meant by phlogiston, because it depended on the scientist using the term and the experiment in question. Phlogiston and its properties were not rigourously defined during its existence. Phlogiston sought to explain what we now understand to be multiple distinct phenomena; that is, combustion and its production of heat and the apparent loss of matter associated with it. Wood had phlogston in it, which was lost on combustion and heat emitted. But wood contained many other 'earths' and left ash behind. Sulphur and carbon would be rich in phlogston, leaving nothing behind as they burned. But they did contain something else, since they produced acid when the combustion products were mixed with water. In that context, phlogiston is effectively "negative oxygen". Phlogiston was also a substance of great 'levity' (and here we go all the way back to Aristotle), explaining why the combustion products would rise and disappear.

Cavendish, upon discovering hydrogen (inflammable air) as its own substance in 1766, thought that it might be pure phlogiston - it burned with a barely visible flame and produced no ash or products (later he would discover that water was produced when burning inflammable air in dephlogistated air), and the gas was a substance of great levity. The identity of phlogiston with hydrogen was shared by Scheele and Kirwan among others, although their theories cannot be reduced to that alone.

Priestley would also term pure oxygen as 'dephlogistated air'. But phlogistated air was CO2. Hydrogen doesn't quite fit with that.

Jean Bodin had already in 1596 observed another mysterious thing. Burning/heating metal turned it to a powder, a calx (metal oxide), which weighed more than the metal had done. While adding carbon to a calx and heating it produces metal again. Boyle, pre-phlogiston, had explained this as fire particles having combined with the metal. This was an apparent challenge to phlogiston theory was handled in various ways (besides claiming it to be in error, of course)

This is not as big as a discrepancy as it might occur to us now that we take Lavoisier's conservation of mass for granted. If phlogiston weighed less than air, then ought things not weigh more if they lost it? Perhaps the loss of mass on burning other matter was simply due to the acids they form? Metal calxes on the other hand are bases, so this can seem quite reasonable. Another idea was that the loss of phlogiston was accompanied by some acid-making substance combining with with things like carbon, but being lost with metals. (let's note here that 'oxygen' means 'acid-making')

Stahl himself acknowleged that metals got heavier as they lost phlogiston and formed a calx, but it's not entirely clear whether he considered phlogiston itself to have mass.

Lomonosov disputed Boyle's interpretation by heating metal in a closed glass vial, finding the total weigh of the vial not to have changed. (However, as opposed to frequent claims, Lomonosov did not draw any general conclusions on the conservation of matter from this) Lomonosov explained the phenomenon through his own theory of gravity. He also had a peculiar mechanical theory of heat of his own, even though he was a phlogistonist (despite claims to the opposite)

The result though, is that phlogiston theory was not much of a scientific theory in the modern sense - even though it became widespread, it didn't have much predictive value. Phlogiston only really comes to the forefront as it comes under scrutiny and attack, around the 1770s-1780s. But at the same time it's a moving target, as I mentioned - the theory developed and changed with Cavendish, Priestley, Kirwan, etc. In Kuhnian terms this was of course a 'crisis', being one of the defining examples in The Structure of Scientific Revolutions. However it has been challenged, not least in recent years, whether phlogiston ought to be considered a paradigm in the first place (e.g. Kim 2008)

The revolution of the Lavoisiers and their close collaborators was not just about throwing out phlogiston, but jettisoning all old alchemical rubbish from chemistry that lacked empirical support. For instance old mystical planetary names and symbols - just say 'copper sulphate' not 'vitriol of venus', much less '🜖♀', not to mention stuff like 'salt of crab's eye' (calcium acetate). Lavoisier was a republican but the revolution he was plotting in 1789 wasn't the one most think about. Nevertheless he did correspond with people he disagreed with, from all over too. (It's rather fascinating when you consider the discovery of oxygen is shared by a English theologist sitting in rural Wiltshire, a French tax farmer in Paris, and a Swedish small-town apothecary. Even if Lavoisier to his shame denied getting Scheele's letter announcing the discovery. A century later it was found in Lavoisier's archives)

So Lavoisier came up with 'caloric theory' instead, separating out heat/light from being any sort of corporeal matter, even if he listed it as an 'element', it lacked mass. The likely reason why he grouped it with elements was that considered caloric to be conserved, as mass was. He describes it as a fluid which can be transferred from one body to another, and has for instance the effect of causing expansion, getting in-between the particles in matter to move them farther apart. However, unlike many phlogistonists or indeed any scientists of his time, Lavoisier was very lucid about the ontological status of his substance. In Traité Élémentaire de Chimie he more or less states that he cannot prove caloric exists, and that it need not actually exist merely that something that causes its described effects does exist. In other words, caloric theory in Lavoisier's view was not actually incompatible with the later kinetic theory of heat (which it's frequently put in opposition to), other than in the conservation aspect. But that aspect was in turn key to Carnot's theories (1820) which were the start of thermodynamics. (Carnot himself would abandon caloric in that sense, having discovered Joule's results before Joule did, but not publish them - likely because he couldn't reconcile it with his earlier ideas. They weren't reconciled until the Second Law of Thermodynamics came about in the 1850s)