The English verb "to sit" can mean either static position, or movement into a sitting position. The addition of "down" makes the movement explicit.
As such, "I sit here" can mean either "I remain here in a static sitting position," or "I move into a sitting position here". The "down" is optional.
Thus, requiring the "down" in the translation for ideülök is incorrect: this "down" should be optional.
It does not need to be "le-" for it to be a motion towards sitting. It could be any one of le/fel/ki/be/át/rá/ide/oda/vissza, and probably a few more. Plus some postpositions. They are not added to "le-" but, rather, they replace it. Usually.
You should really not concentrate on "le-" but instead try to learn all those preverbs, postpositions and locational indicators that have to do with motion to a target. Some of them are part of those groups of three (to-at-from), like "elé, előtt, elől".
"Ide-" is the "to" part of "ide, itt, innen", so it has no problem indicating a sitting down action "to this location".
"Ide" can be both a preverb and a word on its own right, so, yes, you can also say "ide leülök" - "to this location I will sit down".
Maybe, just maybe, this motion to a location thing could be expressed through the verb like this: "I am not going to stand here, I am going to sit here". This is not perfect, either, just an idea.
Actually, I worked quite hard trying to grasp the differences between the various preverbs and suffixes. I think that I am grasping their use in most cases (in a great part thanks to your explanations). If we had ideülök a szekre, I would translate: I sit down here onto the chair. But here we just have ideülök. Without additional information, I am wondering why "we sit down here" is accepted only, and not "we sit here". In your last sentences, you implicitly assume that there is a motion. Hence my question: can ideülök express the fact that I am sitting already here?
No, "ide" in itself, without further anything, is already indicating an action/motion "to here".
"Ide ... a székre" is literally "to here ... onto the chair". So, "a székre" only identifies the target and repeats the indication of a direction.
But it can still be omitted, the other part still carries the motion. Or vice versa, the "ide" part can be omitted, the "székRE" still carries the motion. That motional indication therefore appears twice in "ide ... a székre", but one is sufficient to convey that meaning.
If I am already sitting here on the chair, it is "itt ülök a széken".
The trio of these two words together looks like this:
- to: "ide ... a székre"
- at: "itt ... a széken"
- from: "innen ... a székről"
Either part of any of these pairs already sufficiently determines the "to/at/from" identity.
It depends on what we are emphasizing. In this sentence, the emphasis is more on the verb, not so much on the negation, "nem". What I am doing is not THIS but rather THAT. So the verb itself is in focus, hence it stays intact.
I don't know if you can sense this kind of difference in emphasis between these two sentences in English:
This is NOT an orange.
This is not an ORANGE but an APPLE.
It is quite usual in these "not A but B" type sentences that the emphasis is on "A" and "B" themselves. And when "A" and "B" happen to be verbs, they will also get the emphasis, and stay intact.
If you want to get a bit confused about this, you can read this discussion:
Does "ideáll" only have the purely 'static' meaning of standing here or the 'dynamic' one of achieving the state of standing here?
i.e. is it more "I'm standing here (as I have for the whole recent past)" or "I stand here (for example, stopping walking)".
I'd assume that because "ide" has connotations of motion towards, it's the latter, but I'm not certain.
(In the case of "ideül", it's been clearly explained in these great comments that it's "sitting down" ('dynamic'), rather than "being seated" ('static').)