My problem has been solved, however I don't think the way I did it should be the way everyone else should do it:
I went into the folder of the game and I deleted everything related to CoH / Homecoming / Tequila. (I saved the « keybinds.txt » file on my desktop.)
After that, I double-clicked on Tequila.exe that I saved on another drive. Went on with the instalation. The I got an error telling me it was unable to download an icon. I went in the options and clicked Re-Validate. It apparently solved the problem, cause after that, I clicked Play on the 64-bit version, logged-in, selected the server and my character (wich all were there) and went in the game.
So, like I said, my problem has been solved. However, I don't think my way should be the way everyone else should be cerrecting this. I'm pretty sure that most people are not comfortable enough with computers to do what I did (wich is not a big thing for some of us, but I think it is for a lot of people).
That being said, would it be possible to publish a quick and easy way to fix this for those who are not familiar with computers?
Also, I assume that for those who DID NOT UPDATED THEIR GAME THIS MORNING, the patch will be adjusted in order to not have this problem... (However, at this level, this is beyond my small knowledge of computers... 🙂 )
Thanks in advance to those working on this problem!