How do I change the account in use for Hangouts (Chrome app) (or alternatively how to reset its data to re-log in

You need to change the default account in your multi-account setup.

To do this, sign out of everything (all accounts in all google apps) in Chrome. Then sign in again, in the order you want for priority for extensions and apps.

The first account you sign in as will be default and will ride hangouts extension.

It's a quirk in the multi-login system Chrome and Google use. Really they need a 'set as default' button to save having to this every time you want to switch hangouts accounts.

You now also have the option of adding another user to your browser. Click on your name in the title bar, add a user. Then sign in, and add the chrome plugin. You'll end up with two instances of the plugin.

In my case, we had a link on our calendar events and Slack for the hangout. Turns out there's a URL parameter at the end you can change if needed. https://hangouts.google.com/hangouts/_/example.com/your-conference-name?pli=1&authuser=1

Note the authuser=1 which can be switched, which changes the default account.