It worked! Thanks a lot c.gold!
I did all like you said, except that it didn't take one minute. In fact, it was a lot longer than that. I have roughly 8000 songs on my iPod Touch and to sync them back on it took close to two hours. But it's all fine.
I also had to sync my iPod twice. The first time, after all syncing activity seemed over on iTunes, I unplugged my device but then I noticed that the circular arrows (top left on the screen of my iPod) were still "going 'round". So I figured something went wrong. As a matter of fact, some songs were still randomly missing on my iPod. So I forced it to stop "syncing" by switching it down (holding the top button for a few seconds) and I plugged it back on my computer. Then I resynced it and this time, I waited until the arrows were done spinning on the iPod screen (it happened long after iTunes itself was "done" syncing). And now it's all fine.
I wonder if I'll have to go through all this ordeal everytime I want to add new music on my iPod, but at least, it's a way out of this problem. I'll post here in the next few days to tell you how my next syncing goes.