By the way, there are mappings between these fields. There are ways of doing geometry in which you won’t see any shape or drawing ("you shall not make for yourself a carved image, nor any likeness"), and everything looks exactly like working with numbers and abstract structures.

Mathematics (from Greek: μάθημα, máthēma, 'knowledge, study, learning') includes the study of such topics as quantity (number theory),[1] structure (algebra),[2] space (geometry),[1] and change (analysis).[3][4][5] It has no generally accepted definition.[6][7]

Two points:
It seems to me that the question is not merely semantic, for the simple reason that even you yourself define mathematics as a logical and systematic enterprise. It seems that in your view this is the necessary essence on the basis of which one can describe the common denominator between mathematics and geometry. If you didn’t think so, you presumably could have argued that the basis is something else—for example, psychological.
The second point is that this basis you proposed, the logical one, seems too broad to me, since it includes additional branches besides mathematics and logic—presumably those you mentioned yourself.
From here the question arises: is there a necessary and narrower basis shared only by mathematics and geometry? Here I see two possible answers (actually three, but I’ll ignore the third for now): yes or no. If yes, then it would be necessary to define that basis and then ask again about the difference between the two, this time on the basis of that alone. In that case, maybe we would need to return to the distinction between the different objects to which these two branches are directed.
If not (that is, if there is no shared basis besides the logical structure you mentioned), then it is in any case easier and more correct (and perhaps also more practical) to divide between the fields—just as you yourself distinguished between them and between them and the other branches (trig, etc.).
Also, as I understand it, your remark that there are areas within geometry that look like mathematics should be interpreted a bit differently: perhaps the attempt to call them geometry is not justified (even if useful)?

One can return and ask what the point of these definitions is. In my opinion, there are two reasons: the first is purely theoretical—definitions, or at least verbal descriptions, are the main (or only) working tools we have in philosophy. It is simply a question of truth, and that has, at least in my view, intrinsic value in itself.
And there is a somewhat more practical answer: looking at fields of knowledge from a misleading or partial angle may block the development of those fields (I’m not claiming this is necessary, of course). I won’t expand on that right now.

These aren’t areas within geometry, but rather different ways of treating geometry itself. And that means the difference from other branches of mathematics (like those that deal with numbers) is not essential.
The definition I gave for mathematics is, in my opinion, the definition. Anything that falls under it is mathematics.

By the way, would you say that your definition of mathematics falls under logicism? Because if so, it’s a pretty extreme logicism.