Well, Raquel. I think I'm going to confuse you more! As in the example I gave:
Also, to avoid the present continuous "am forgetting", we use the simple:
"Ince has developed into a player of substance and people forget he is still only 24."
...we do use present simple to express something happening now to avoid the awkward (or impossible) continuous (as we do with many verbs):
I didn't realize that before but I SEE now.
I'd forgotten about that but now I REMEMBER.
Your American friends are right and this usage is heard in UK too but I detect a difference between:
"What I did yesterday, you ask? I forget" and:
Me: "I 'saw'...
Carmen: "Yes, yes. I forget. I 'saw' Sofía."
In the first, there is an obvious reference to the present; she didn't forget yesterday; she's forgotten now so the simple or present perfect also possible. In the second, Carmen refers to the past mistake she's just made; I detected no present meaning so "I forgot". Epilogue:
Was Carmen thinking "Oh, no, I always forget to use my past tenses!"? If so, I concede and admit Carmen could have said "I forget" and have been correct.
I think the key is does the action refer to present or past?
Even the Americans and Jane Austen would admit you must say:
"Did you remember to close the door?"
"Oh, no. I forgot!"
Present reference: I forget, I've forgotten (in some set expressions "I'm forgetting")
Past reference: I forgot.
Hope this helps.