I have a hard time reading/understanding these names (that can be true or false):
on_check_updated_in_callback
on_check_closed_in_callback
I don't know how to read them :) making sense of what they do while reading these names. I can't even suggest better names :)
Not sure it is something with English grammar/wording, or me just being stuck or used to thing ending with _callback being actually a callback/function (we have on_close_ges = {...}, on_tap_close_callback = function(arg, ges, footnote_height) , on_success_callback(entered_preset_name).)

If we can't find better names, I think they deserve some comment in the TouchMenuItem declaration.

"on check" is a prefix, this is about items with checkmarks only, "on check" means "on tap".
Then, the menu is updated in the callback, or the menu is closed in the callback.

This flag is placed right before the callback function, to be clearer that it is about the following callback.

    on_check_updated_in_callback = nil, -- set to true for item with checkmark if its callback updates menu
    on_check_closed_in_callback = nil, -- set to true for item with checkmark if its callback closes menu

I'm still bothered by the on_....._callback, where everywhere we would expect it to be a function.
What about ?:

callback_updates_menu = true,
callback_closes_menu = true,

on_check came from the former name no_refresh_on_check.
I think the names should show that the flags are applicable only for items with a checkmark.

check_callback_updates_menu = true,
check_callback_closes_menu = true,

checkmark_callback_updates_menu = true,
checkmark_callback_closes_menu = true,

callback_updates_menu_on_check = true,
callback_closes_menu_on_check = true,

Starting the things with on_ or ending them with _callback is what I'd like to avoid.

Okay