Is the meaning of the word "semblance" closer to that of "fake" or that of "illusion"? I mean, does it have the negative connotations that "fake" or "counterfeit" have, or is it something that can be more innocent, maybe even positive, like "illusion"? (Or am I just completely misunderstanding the meaning of "semblance"?)
semblance has to do with a things appearance resembling something else whose actual substance is known while acknowledging that this resemblance does not guarantee the same substance. It merely hints at a possibility not yet proven but made to seem to be possible.
It doesn't actually have a negative or positive connotation. The word embodies the expression "You can't judge a book by it's cover".
Bad cover, good book. Good cover, bad book. Good cover, good book. Bad cover, bad book.
The only connotation really comes from your own expectations.
For example if I say, "I hope the book bears some semblance to its cover" I haven't even told you whether I think the cover is good or bad. I've only made clear that I hope the cover reflects the contents while clearly acknowledging that it might not.
Semblance can be used in the way you suggest but it need not be used in that way. The meaning often hinges on the adjective:
"mere semblance..." would indicate that the thing is all show and no substance, whereas "some semblance" would indicate that the thing bears at least some resemblance to that which is desired. Compare "some semblance of decency".