Is the 12-word phrase app-specific? I'm new to Bitcoin and trying to understand it, step by step. I tried to make a wallet using Electrum. This gave me 12 words which seem to allow me to regenerate my wallet's private key. When trying to use this 12 words in Exodus (another wallet app) it says those 12 words are invalid. So, I figured these 12 words might be an app internal thing...
Seeing Chris Chen's comment here seems to confirm this.
So, the question would be: if I happen to get my 12-words compromised and someone gets to know them, would I then be still safe as long as this person does not figure out which wallet app I used?
No your funds would still be at risk. There are only a few popular wallets that use seed phrases.
Most wallets use BIP39(https://en.bitcoin.it/wiki/Seed_phrase) but there are some that use different versions or no seed words at all.
Electrum does not ouse BIP39, see their docs explaining why. https://electrum.readthedocs.io/en/latest/seedphrase.html
It would be extremely easy for an attacker to try the most popular versions and steal your funds.