I just grabbed your test code, modified it to call system.api.list, and tried running it on my localhost in 2 ways:

• hit localhost and it succeeded
• hit the same endpoint on the community site and it failed

So have you any clue on how I can handle this problem ?

Well, that doesn't establish were the problem is. I don't have time right now to determine why that jQuery code is failing on the call to the community site (and it's not an empty string for it - it is calling the error callback).