This is why it's important to have a good mentor who knows good things about crypto or Bitcoin. If you start learning wrong from the beginning, you will have this wrong thinking for a while before someone corrects you. You may start debating with other people without knowing that you are wrong. It is essential to make sure that what you have learned so far is not incorrect.
How do you decide whom to trust and why? Bitcoin was designed to avoid having to trust other parties as long as you own and hold your private keys in your own wallet.
I agree with OP that it's beneficial to know as much as possible how a wallet works and what a crypto wallet actually is. Your coins are not in your wallet, only your private keys are in your non-custodial wallet. Coins only "live" in the blockchain and the private keys give you the ability to move coins around. That's a very short answer to it. The devil lies in details...
In custodial wallets, like those on exchanges, you don't have and own the private keys for your coins, therefore: not your keys, not your coins. With a usually closed-source custodial wallet the only thing you have is a promise of the wallet provider to eventually give you the coins that you're promised to "own" (you own them only at the moment when you've successfully transfered them into your own custody!).
Closed-source wallets, regardless of non-custodial or custodial type, always have the trust issue that you have to hope they work properly and don't screw up. You can't verify what a closed-source wallet does in its code. Why do you trust the wallet provider, based on what exactly??
Verifiable open-source wallets on the other hand show their code to those who can read and understand what's going on. That doesn't make them more secure per se and automatically. But they are verifiable by others and hopefully some knowledgeable code readers will spot flaws and suggest fixes. Open-source also allows participation and improvements by other contributors.
Personally, I would never choose a closed-source wallet for my coins, maybe for 2-figure fiat equivalents only if I had to. And because I don't care about shitcoins/shittokens, I have enough options of reputed open-source wallets (Bitcoin Core, Electrum, Sparrow, BitBox02, Foundation Devices Passport).