On Drupal 6, how can I let registered users edit their own content and also allow them to edit an already existing admin content (just one content type) and then when saved, the user takes ownership of that node?
You can give the 'Authenticated user' role permission to 'Edit any
<content type> content'. If you configure that content type to create a new revision by default, the user will be registered as the author of that revision (and if you don't give them the 'Administer nodes' permission, they can't uncheck the option to make a new revision). The original author of the node will however still be listed. I can't think of an easy way to change that automatically. But the users can already edit all nodes of that type anyway, so it would only be relevant if you use the node author for some listing or so - in that case, you can probably use Rules to do assign latest revision author as node author.
Edit: OK, so in that case this doesn't help. You can investigate if something along these lines works (I haven't done something like this so I'm not sure it's possible without coding):