User Permissions

You don't need groups for this purpose.
When "Users can see data; can edit their own data" security mode is in use UserID is added to each new record automatically.

You don't need groups for this purpose.

Ok, but how do I add a new record as a newly created user? The add button does not appear.

Make sure you enabled the Add page.

The add page is there.
Another thing I noticed is that I cannot log in with my admin ID, which is in the permissions, and works on other projects that are not using the "Users can see data" option.
If I enter the add page manually into the address bar it then tells me my session has expired, please login again.

It looks like you use User Group Permissions and new user belongs to the group that don't have access to the Add page.

Proceed to Security->User Group Permissions and enable access to the Add page.

Sorry to be thick-headed, but what group is assigned to new users under this feature? How do I link this feature to a specific user group?

It all depends on GroupID field default value. Via default value you can control what group new user belongs to.
Also you can setup a Default group. All unassigned users will fall into this group.

Thanks, I am almost there now! I changed the permissions to the default group and had a successful test. I made sure that guests did not have these permissions, so that's good.
What I need to do now is understand how I recover my admin access to this project. When I try to login with my admin ID it does not accept it.

Make sure admin record is still in Users table.
You can also post your application to Demo Account and send the URL to support@xlinesoft.com.

Ok, I figured this out. I was using a user name that is the same as the group name, which worked fine for older projects that did not use User Login. For whatever reason it doesn't work here, but when I add a new user to the admin group with a name that is different than the admin group name, it does work.