In English, if you say "I forget X," it means that you still don't recall the information in the present. For example, "I forget my first phone number," you presumably forgot it before this moment, but saying, "I forget" conveys that you haven't remembered in the meantime.
If you use the past tense, "I forgot," it leaves open the possibility that you remembered the information later. For example, "I ran into Jake at the mall, and I was about to say hi, but I forgot his name." The speaker forgot the information in the past but has since remembered.
Hope this helps!
I think everything you've said is correct; I just wanted to add the idea of an habitual aspect.
EDIT: found a reference: https://en.wikipedia.org/wiki/English_markers_of_habitual_as...
The meme-famous Oscar Gable quote, "They don’t think it be like it is, but it do," is an example where some vernacular forms of English emphasize it more. Here, "be" is the habitual tense of "to be." It's not "bad" English. It's a tense you're not recognizing.
Things like forgetting and winning would seem to be instantaneous things that don’t ever happen “now” only in the past or future.
“I forget” means something like “I have forgotten at some point and continue to not remember” whereas “I forgot” means “I did forget” without reporting anything on your current state of remembering
“I forget” could also mean “forgetting is a thing i often do and expect to continue to do”
I’ve only learned a few other languages, but none of them have simpler sets of tenses than English (most of them have more differences in conjugation/inflection to reflect them, but also more regularity; English also has a fair number of idioms in which the semantics as far as time don't necessarily match what is normal for the syntax of the tense, but I’m not sure its unusually dense with those.)
EDIT: But:
> “I forget” means something like “I have forgotten at some point and continue to not remember” whereas “I forgot” means “I did forget” without reporting anything on your current state of remembering
If you think of “to forget” as equivalent of “to fail to remember something which one once knew” (which I think is the best understanding of that verb), that’s pretty much just what you’d expect from the present and past tenses, respectively.
> b. To fail to recall to mind; not to recollect.
> 1787 ‘G. Gambado’ Acad. Horsemen 12 He says much the same of rabbits and onions, but I forget how he brings that to bear.
> 1847 F. Marryat Children of New Forest II. i. 3 I forget the sign [of the inn].
> I forget what kind of computer it was; an early type of PC that belonged to one of the professors.
As a ESOL speaker, it always baffled me how native English speakers would sometimes use present tense "forget" to write about the past. Why is that? The only other time I see same oddity is with "win". I have seen this in enough of high-profile text to assume they're not just typos.