If I bought bitcoin in the early days in some exchange for $10, would it have been lost if I kept it in the exchange?
You are correct.
Leaving your money in the hands of a third party such as an exchange means two things

so far as the Bitcoin network is concerned, you own no money at all.
when that business fails, your money is usually lost. You might get some back if there is a police investigation and/or some bankruptcy process - but you shouldn't rely on this.

Transferring the Bitcoin amount to the sole control of your own Bitcoin wallet means that your money is safe, so long as

you keep a copy of the private key(s) or seed-phrase somewhere safe, or make regular backups and remember any password.
your computer isn't hacked or infected.