Added a screenshot to illustrate the visual difference:

Have no idea regarding the question standard vs. custom though.

As a part of this it would make sense to term it as "Configure" and "Delete" instead of "Edit" and "Delete" (see backdrop/backdrop-issues#771)

Easy part is renaming Edit to Configure. We don't think there's an easy way to change the buttons into the drop-down. The button is javascript driven.

There's a similar issue on the Manage Display side:

Where to put that Delete link?

One potential (perhaps) is to change the operations to the dropdown and then load the configuration options in a modal instead of an AJAX-inline element. (Similar to how the "Configure" drop button works on a block on a "Manage blocks" layout).

@jenlampton I created a replacement PR #37 of this since I had moved code around to separate out the API and the old PR wasn't applying cleanly anymore.

Since this PR is basically borrowing your work, I'm happy if you want to take it over to get it over the finish line or if you want to work together on it. I stopped when trying to get a title for the configure form so that someone knew which field group was being edited.

@herbdool just indicated in Zulip that this might be a blocker to getting Field Groups into core.

@stpaultim personally I'm ok if it goes in without this, but others disagree