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For the most part, Reptilia just isn't really used as a formal taxon anymore. It may sometimes be used as a convenient grouping of more basal or less metabolically active amniotes, but in this way it usually applied to extant or recent groups (i.e., the classic collection of lepidosauria, turtles, and crocodilians) in which case there's no need to specify the inclusion or exclusion of early synapsids. The definition of reptilia as basically synonymous with sauropsida was an attempt to sort of preserve the term as a proper monophyletic clade, but in my experience researchers mostly refer to sauropsida instead to avoid any ambiguity.

We're all just fish anyway.

Seriously though, synapsids and sauropsids share a common ancestor but are two distinct monophyletic sister lineages. Synapsids gave rise to mammals and sauropsids to reptiles (including birds which are just highly derived reptiles).

Your question just seems to be one of nomenclature rather than taxonomy/systematics though. Reptiles are reptiles from cultural carryover, even though classically the definition is paraphyletic (by excluding birds it doesn't include all descendant species of a common ancestor); in modern systematics it includes birds and makes a single clade. Mammals are a distinct monophyletic grouping and so remain a valid clade.

Changing amniote to reptile and synapsids to "mammal like reptiles" and sauropsids to "lizard like reptiles" doesn't add any clarity to things, and because lizards/birds/snakes/turtles are all fairly distinct groupings on their own it actually muddies the definition some.

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Historically reptiles used to mean all sprawling or semi-erect amniotes. So all amniotes besides mammals and dinosaurs (which are erect)

Some people want it to be a monophyletic taxon, but all the relevant ones are already taken. They could have chosen that reptile means amniotes, but they choose sauropsids. In practice, reptile isn't used as a monophyletic taxon. When it is used, most people mean the historical definition.

Amniotes and sauropsids are already clear terms and don't need to be replaced.

Like u/DrDirtPhD said, it wouldn't clear anything up. If we assigned all amniotes as reptiles that would make us reptiles too (as it is, people have a problem with view birds as reptiles).

But also, the choices here are to help break the older "ladder thinking" and similar biases in biology.

As such, calling synapsids "mammal-like reptiles" and reptiles "bird-like" or "lizard-like" reptiles has a heavy implication of directionality to it. Evolution doesn't move towards some goal. Birds and mmmals aren't some pinnacle of vertebrate or amniote evolution. We all exist, some survive, some die. We all seem to have our niches (more or less, life changes, and humans are... complicated). But none are inherently "better" than anything else because that isn't scientific, and trying to force a zebra to live like a lizard wouldn't work, just as an example.

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Huh. I never thought about it like that. But when you think about it I guess you're kinda right because saying we're descended from mammal-like reptiles may imply to people with the ladder thinking that reptiles are just "less evolved", when they're not.

Still would've been nice to be able to say I was descended from a dinosaur though lol.

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Exactly. Takes the implications right out of it. Sprawling gaits and ectothermy work well for some, no need to change it, doesn't mean they're "primitive" just means they're undergoing different selective pressures!

lol, yes. But. to say we come from such a beautifully complex lineage is pretty cool too. Enigmatic in some cases. More fun that way, imo.

Are we sure Synapsids and Sauropsids are distinct clades? So their common ancestor is neither?

The diapsid infratemporal fenestra is homologous to the synapsid single temporal fenestra, so this asserts that the amniote line first developed a single fenestra (the [infra]temporal fenestra) and then later one lineage gained another (the supratemporal fenestra), with this lineage becoming the diapsids while the other being synapsids.

With this in mind, why are the early amniotes who first developed a fenestra not considered synapsids? Is it to maintain synapsida and diapsida as monophyletic that we only consider them synapsids a while after their defining trait developed?

Or, rather, why are diapsida not considered a clade of synapsida given that the common ancestor of both lines already possessed the infratemporal fenestra? It would seem sensible to me to put them both in a clade specifying a single fenestra ("monapsid"?) with the synapsids just being those more closely related to [insert synapsid here] than to diapsids.

I haven't encountered the same reality in the herpetological circles I'm part of. Most researchers continue to use reptilia and reptile as terms, either including avian reptiles or excluding them with the understanding that they are technically a part of this group. You will find many herpetologists use reptile in the same context that sauropsid would be employed, but don't bother being extremely technically correct, given that everyone understands the intent.