I also read people advising you to keep your password through a third-party management online, I don't encourage that, I've heard enough of password compromise and that may be one of the reasons, just don't trust the internet for anything.
A password manager isn't necessarily an online service, though such exist and make cross-device synchronization a lot easier. If such online password managers are made with security in mind, at least at the very basic level, they only store and sync an encrypted password vault which is only decrypted on your local device. Of course you have to trust the service that only you have the decryption key.
You are right and I know all that, but I can't advise against such an app/service if I do not know one or two about it, some services are beyond what we think. Aside from what I read online, I've also been a victim of the Google Password Manager. It was bad to the point that on successful login to all the websites I used it for, there was always a popup warning me to change my password as it had been breached online.
I only use Google Password Manager to save my password for easy access. From that time, I've vowed not to use it again no matter what anyone tells me. The best way to keep your password is either to memorise it or safely keep it manually where no one would have access and where internet cannot reach.