As you can probably tell, from my understanding (possibly wrong connection) I think of the word 'to forge' in the context of actually trying to get access to somewhere or something with fake documents. When the intention or the consequence of using such document is more for monetary or material gain, I tend to think of it as fake.
I suppose dictionary definition does allow for both usage, i.e. to forge is to make a fake resemblance of something.. etc. I didn't think not knowing accurate definition of a word warranted downvotes but oh well.