Re-define lilypond's `format-mark-box-alphabet`

Question

Say I wanted to re-define the format-mark-box-alphabet using a custom scm command.
How can this be done?

Here is a MWE where I would want both marks to be identical, with the double outline.

\version "2.24.1"

% Custom definition of the marker command
#(define-markup-command (format-mark-box-alphabet layout props text) (markup?)
  "Draw a double box around text."
  (interpret-markup layout props
    (markup #:override '(box-padding . 0.4) #:box
            #:override '(box-padding . 0.6) #:box text)))

\score {
  \new Staff {
    \set Score.rehearsalMarkFormatter = #format-mark-box-alphabet
    a2 a4 b
    \mark \default
    c'1 c'
    \mark \markup \format-mark-box-alphabet A
    a
    c'
  }
}

Current output:

Desired output:

Answer 1

I think this is an XY question. Redefining a built-in procedure is called monkey-patching, and it is not advised, neither in LilyPond nor in most programming languages that allow it. Instead, you want to define your own rehearsal mark formatter format-mark-double-box-alphabet, which can be done like so (I've also used \etc for the markup command):

\version "2.24.1"

\markup double-box =
\markup \override #'(box-padding . 0.4) \box \override #'(box-padding . 0.6) \box \etc

#(define (format-mark-double-box-alphabet number context)
   #{ \markup \double-box #(format-mark-alphabet number context) #})

\score {
  \new Staff {
    \set Score.rehearsalMarkFormatter = #format-mark-double-box-alphabet
    a2 a4 b
    \mark \default
    c'1 c'
    \mark \default
    a
    c'
  }
}

Namely, the mark formatter is a Scheme procedure that takes a mark number and a context, and returns a markup. In this case, it's enough to use the built-in format-mark-alphabet procedure to get a letter from the number, then double-box it.

Answer 2

I fully support Jeans answer, but I want to fill in some details about why your version does not work. The reason is not that you are trying to redefine built-in procedures (which you can totally do, but usually would be a bad idea because in later code it becomes hard to understand if we are using the internal procedure or a modified one).

The reason for this is that you define your procedure only as markup command. Now, markup commands are a bit special in Lilypond, as they are defined in a separate scope (also they works slightly differently in when they are actually evaluated). So when you do

#(define-markup-command (test layout props) ()
   empty-stencil)

#(display test)

you will actually get an error, as guile won’t find the binding for test. In your case this means that while you define a markup command format-mark-box-alphabet, but this does not change the binding for format-mark-box-alphabet, so you are still using the old functions. Needless to say, if it actually tried to use the markup function it would encounter some issues, as Score.rehearsalMarkFormatter is not expecting a markup command, but a function with a different interface, as shown in Jean’s answer.