That is what I do. A Text file that is named to fit right into a directory of other system files, in a weird sub sub folder. I increment the file name so I have an idea of when it was edited. Then I use a app on my iPhone called Air Disk Pro that works like a WI-FI thumbdrive. It is a secure password protected app, and I can store any type of file from my computer in there. I keep a copy of the password list in there for occasions were I am not at home. I store copies of important files in there.

Comes down to levels of comfort I guess. Password protected doesnt mean much to many. Encrypted? With what? How strong? Good point about trusting a 3rd party but you have to weigh not just the storage function but the complexity of the passwords generated. And its encrypted locally. :)

+1 and its the one I recommend to family and friends.
The other ones that are hosted (cloud based) are just targets waiting to be hacked.

The site is a little busy wrt content for my level of understanding. This is not cloud based and totally under my control locally (my hard drive)?

Absolutely, this is the one solution that has been truly audited by 3rd parties and is the one that physically lives in your PC of choice. (no cloud and no fees)

Maybe install it on the PC that is not connected to the Internet if you want to be super-safe.

Download from here:
https://keepass.info/download.html

And choose the "KeePass 2.40 Installer for Windows (2.40)"

That one will install automatically with no intervention and will automatically check for new versions (if you decide to allow of course).

All the other ones are more specific to power users