I want to associate the user's uid with their record(s) automatically. In the .cfg file should I name the (hidden) field uid or something else to avoid conflict?

Naming the field Userid as it is in the default settings will keep the record associated with the particular user. You just need to keep in mind the Userid field number needs to be identified in the auth_user_field setting a little further below in the config file.

Userid => [9, 'alpha', -2, 15, 0, '', '']

$auth_user_field = 9;

Karen, thanks.

But I'm confused: what's the relationship between Userid, userid, uid and $db_uid - all of which appear somewhere in the scripts as supplied?

I'm not a 'coder' so can't give you precise details although I'll try my best to explain how I've seen the terms used & integrated within the scripts and usage.

Userid => fieldname & placeholder within the db
userid => don't know although I have seen discussions here in the forums on the Upper/Lower case versions of the word.
uid => This one I've seen used for passing the Userid in URL's etc - for example
db.cgi?uid=default
$db_uid => Used within the coding to access/track the Userid as entered in the database.