May be stay simple like every where else - and consistent with "Do not show this book cover...":

Do not show book cover >
  Do not show book cover on finished books
  Do not show book cover in file browser

(Or "Ignore" instead of "Do not show" which was my first idea when reading "Hide..." - we're not really hiding it.)

(Feels a bit excessive options, we rarely stay on a finished book or in File browser - except may be as a way to explicitely not wanting to let the device with a cover, just switching to FM before suspend.)

the part about closing from FM is something I have wanted for a bit, since I read in different devices and usually use a different one for every new book. the other bit is taken from #11080. I'm not sure about the Ignore book cover rules though

Maybe something like

Show book cover:

• always
• not when finished
• not in file browser

Show book cover

• always
• only when reading
• only when in document

Choose to programatically not show feels a bit too verbose, we have "programmatically" nowhere else, which is not surprising for a program which does it all programmatically :)
Decide to show or not book covers ... or something more straigthfoward.

Maybe something like

Choose when to show book covers on the lock screen.

How about this: ;)

Forsooth, by digital alchemy’s might,

Cloak book facades from prying sight;

When device locked or tale complete,

Let covers vanish, discreet and neat.

A distinct lack of spacing. But yes, "choose to programmatically not show" must go, I was just too distracted by the regular text. :-)

fixed, the spacing ;)

What is this stored in doc_settings ?
doc_settings:isTrue("exclude_screensaver") is in it because it's a per-book decision.
Your check can just be dynamic in here, it does not need to store anything anywhere, doesn't it?

I suppose not. but finished is also a per book decision.

Not familiar with all the screensaver modes, but you wrote "both the cover and sleep screen message": does the message in question only shows when screensaver_type=="cover" ? or are there some other modes which display some info about the book as a message over a black screen or random image - and does this setting apply then?

does the message in question only shows when screensaver_type=="cover" ?

nope, but with all due respect to those who don't use the book cover. They are all, wrong ;). This PR exclusively acts on book covers, everyone else stay the same. the sleep screen message can be plastered on top of anything.

Missing screensaver_exclude_on_hold_books ?

yeah, apparently in rushing to do the on hold before leaving, standards were lowered.

All this logic is tedious to read - and because of that I think there is a bug (see other comment).
You should write it simpler and more logical

excluded = doc_settings:isTrue("exclude_screensaver")
if not excluded then
    if not ui and G_reader_settings:isTrue("screensaver_hide_cover_in_filemanager") then
        excluded = true
    else if ui then
        if  G_reader_settings:isTrue("screensaver_exclude_finished_books") or G_reader_settings:isTrue("screensaver_exclude_on_hold_books") then
            -- do your summary cuisine and checks and set exclude = true
        end
    end
end
if excluded then
    self.show_message = false
end

Looks like if I have G_reader_settings:isTrue("screensaver_exclude_on_hold_books") and the book is finished
or G_reader_settings:isTrue("screensaver_exclude_finished_books") and the book is on hold,
you will exclude that book cover.

good eye.

Please re-order them so hide_cover_in_filemanager doesn't end in the middle of the other more-related 2 (put fm first, so the exclude_ ones are visually aligned).

Looks like these 2 big similar branches can be merged ? (more checks even if already excluded, but it avoids duplicating the whole thing):

            local book_on_hold = book_summary and book_summary.status == "abandoned"
            if not ui and G_reader_settings:isTrue("screensaver_hide_cover_in_filemanager") then
                excluded = true
            elseif G_reader_settings:isTrue("screensaver_exclude_finished_books") and book_finished then
                excluded = true
            elseif G_reader_settings:isTrue("screensaver_exclude_on_hold_books") and book_on_hold then
                excluded = true
            end
            if excluded then
                self.show_message = false
            end

These local variables are badly named:
exclude_finished_books = G_reader_settings:isTrue("screensaver_exclude_finished_books") and book_finished
The G_reader_setting things is plural, but and book_finished have it apply on the current book, so the local var can't be plural.

The blank line is not needed, as the whole section is about the same thing.

And I'm sure you can do shorter than adding 20 lines and you don't need that many branches.

Good for what's above that.
For these 2 branches, it looks to me they are just logically equivalent to one and clearer:

if should_exclude_book then
    excluded = true
    self.show_message = false -- get rid of the message
end

I hate you ;)

I know - but it warms the heart to hear it. I hate you too :)