plugins – Custom Post Type action on post update

I’m trying to trigger a specific action when a custom post type that I’ve created is created/updated.

Ideally, I could run the same function/process whether the post is:
Created using the New Post screen in the WP Admin
Updated using the Edit Post screen in the WP Admin
Updated using the Bulk Edit function that I’ve created
Updated using a Quick Edit

I’ve reviewed and experimented a bit with save_post_{custom_post_type}, pre_post_update, handle_bulk_actions-edit-{custom_post_type}. Save_post would work, except on a bulk update, it appears it only gets called once, with one post. handle_bulk_actions-edit would work, and I can iterate over the IDs provided, but that excludes editing posts manually on the post edit screen, etc.

I also looked at update_{$meta_type}_metadata as that hooks on specific metadata fields, but that seems like a hack given I really need it on the post level.

Is there one hook that gets called on ANY post add/edit that I can use regardless of the source of the edit (post edit screen, bulk edit, quick edit) ? Or what is the best way to handle?