Thank you
So concretely: how can I save a few thousands $ worth of BTC without having to rely on an online service trust?
You'd use a wallet like electrum and bitcoin core. In both of those cases it's highly unlikely you'll get scammed by their developers and you do own the private keys.
Should I save the private keys? OR the seed "big number"? OR the seed list of words on a piece of paper?
What software to use?
The seed is all that is needed. If you use a bip39 wallet you may also get a master private key that yoy can also store in case of transcription/data loss errors.
Let's say I want to use 2 pieces of paper that will be in 2 locations. What software or what to use to generate the words that I'll write on those pieces of paper? I can't use Electrum because it was said that only Electrum software can generate the "big number" from the list of words. So if Electrum disappeared my piece of paper would be then useless because I could not use any software to convert the list of words into the big number and then into the private keys.
Electrum is a piece of software, it cannot disappear. There is no master deactivation switch in the developers basements. As long as you keep a copy of your seed, you're good. Should electrum not be accessible, there are achieved versions of it stored. There is also the connectivity issue as it is an spv Waller and is a bit irregular compared to regular nodes. In this case, you can import your seeds into bitcoin core.
And when I open Electrum it asks me to select a file. What is this file?
Did you download it from electrum.org? If so, just input a random name if it already says "Default_Wallet". Keep it that way for now.
My goal would be to have those few words on paper and only this paper is needed to access my BTC.
And this paper would be the ONLY way to access my BTC and being able to send them to someone, or to check the balance.
Would that work?
I just don't want any technical hack to be possible, except finding the list of word and knowing what is it, etc...
Yes that would work. Once your words are written down and stored safely you should be good to go. Make a couple of copies of the seed though.
When you say "make a couple of copies of the seed". Do you mean write down the list of words on several pieces of papers placed in different locations?
About the file that is created by Electrum. Do I have to save this file? Or delete it? Does it contain the keys? Or anything else?