How can I give permission to authenticated users to edit the content type 'ABC'?
Use Devel module, especially it's Node Acces tool:

Devel Node Access (DNA)
View the node access entries for the node(s) that are shown on a page. Essential for developers of node access modules and useful for site admins in debugging problems with those modules.

It will tell you exactly what permission is missing.
To debug for particular existing user, consider adding Masquerade to the mix:

The masquerade module is designed as a tool for site designers and site administrators. It allows a user with the right permissions to switch users.

That way you will be able to see DNA report just as the exact user who encountered problem.
What you should see is 2 lists:


If these look about right
Go to Home ⏩ Administration ⏩ Configuration ⏩ Content authoring and make sure your user has the proper text format permissions: