You want to pin down absolute definitions of idealism vs materialism

I want rigour in this stuff, rather than operating on vibes.

Also, I am myself a mathematical Platonist, meaning that I am an ontological idealist myself, and, given how many other socialists both at least claim to subscribe to materialism (which, I would argue, is not always a true claim) and at least claim that Marxism and idealism have significant incompatibilities (which I have not managed to encounter so far), I'd rather resolve this lack of coherence. Either my understanding is incorrect, or a lot of other people are being incorrect. I am fine with the matter being resolved with me being proven incorrect, but so far people have not managed to bring up any relevant incompatibilities.

but the precise meanings of these words are not agreed by all thinkers if they are consciously defined at all

That does not mean that we should avoid defining terms or explain understandings of words. Furthermore, a person can be aware of multiple incompatible linguistic frameworks and try to understand something by attempting to apply each of them. In particular, I brought up the fact that I am aware of multiple definitions for the terms 'idealism' and 'materialism'.
If one refuses to explain what they mean by their words, then they should not expect to be understood, I would also argue.

So what definition of idealism are you applying?

I provided relevant explanations elsewhere in this tree of comments, but the one that I consider to be a 'better' understanding of the word 'idealism' is one that characterises idealist schools of thought as positing that non-material stuff (not necessarily mental non-material stuff) has primacy over material stuff.

How can this be? Marx wrote a bunch of polemics against idealism

Well, just because somebody says something doesn't mean that they are correct. This might seem unwarrantedly harsh, but we do know that Marxist thinkers (obviously, not just them, but only they are relevant here) did not always make tested claims. Some of those claims were tested after being put into works, and some are yet to be tested (like Lenin's anti-parliamentarism from, IIRC, State and Revolution).

IIRC, Marx tried to define idealist schools of thought as positing that mental stuff has some sort of primacy over matter. That definition is bad at least because, according to it, schools of thought like Platonism (and its offshoots) and most variations of religious idealism - famous examples of idealist schools of thought - are not idealist schools of thought, which is silly.
I do not currently have time to delve into those works, as I have thousands of pages of dense reading material to go through that are much more important for me right now.

So, if there are incompatibilities between idealism and Marxism (however you understand what Marxism is), I'm all ears.

Are you defining Marxism as the school that emerged after Marx, or Marx himself?

I am making rather broad strokes here, but I'm pretty sure that what most people here would understand as Marxism doesn't actually have significant incompatibilities with idealism.