@ David: You got it right. The less saturated photo (titled 2.JPG) is the wrong version and it's how it appears in some software and websites. The one named "1.JPG" is how it should be viewed.
As far as I can remember, the colour profile box has always been checked. It is checked right now as well. There is probably something that I do wrong for some reason.
I had a look at the first link you sent. I use Chrome 35.0.1916.153 and I learnt that it doesn't support ICC v4 profiles. Maybe that's why I have so much trouble with warn colours. I will have a deeper look at the second link this week. I had a look at it quickly and it seems to represent my problem.
@ Terrie: The box is checked. I tried the "Save for web & design" option and realized that even there the colours look wrong. It's by default set to "Monitor color", but when I switch it to "Use document profile" it looks like what it's supposed to. However, if I put it on my personal website or social medias, it still looks wrong.
I'll keep you both updated if I manage to solve the problem!