Improve options and docs of M-x command completion

* lisp/simple.el (read-extended-command-predicate): Expand the
doc string.  Add 2 more selectable values.
(command-completion-using-modes-and-keymaps-p): New function.
(execute-extended-command): Mention
'read-extended-command-predicate' in the doc string.  (Bug#60645)

lisp/simple.el

@@ -2207,15 +2207,39 @@ to get different commands to edit and resubmit."
   "Predicate to use to determine which commands to include when completing.
 If it's nil, include all the commands.
 If it's a function, it will be called with two parameters: the
-symbol of the command and a buffer.  The predicate should return
-non-nil if the command should be present when doing \\`M-x TAB'
-in that buffer."
+symbol of the command and the current buffer.  The predicate should
+return non-nil if the command should be considered as a completion
+candidate for \\`M-x' in that buffer.
+
+Several predicate functions suitable for various optional behaviors
+are available:
+
+  `command-completion-default-include-p'
+         This excludes from completion candidates those commands
+         which have been marked specific to modes other than the
+         current buffer's mode.  Commands that are not specific
+         to any mode are included.
+
+  `command-completion-using-modes-p'
+         This includes in completion candidates only commands
+         marked as specific to the current buffer's mode.
+
+  `command-completion-using-modes-and-keymaps-p'
+         This includes commands marked as specific to the current
+         buffer's modes and commands that have keybindings in the
+         current buffer's active local keymaps.  It also includes
+         several commands, like Cuztomize commands, which should
+         always be avaliable."
   :version "28.1"
   :group 'completion
   :type '(choice (const :tag "Don't exclude any commands" nil)
                  (const :tag "Exclude commands irrelevant to current buffer's mode"
                         command-completion-default-include-p)
-                 (function :tag "Other function")))
+                 (const :tag "Include only commands relevant to current buffer's mode"
+                        command-completion-using-modes-p)
+                 (const :tag "Commands relevant to current buffer's mode or bound in its keymaps"
+                        command-completion-using-modes-and-keymaps-p)
+                 (function :tag "Other predicate function")))
 
 (defun execute-extended-command-cycle ()
   "Choose the next version of the extended command predicates.
@@ -2401,6 +2425,35 @@ or (if one of MODES is a minor mode), if it is switched on in BUFFER."
                         #'eq)
       (seq-intersection modes global-minor-modes #'eq)))
 
+(defun command-completion-using-modes-and-keymaps-p (symbol buffer)
+  "Return non-nil if SYMBOL is marked for BUFFER's mode or bound in its keymaps."
+  (with-current-buffer buffer
+      (let ((keymaps
+             ;; The major mode's keymap and any active minor modes.
+             (nconc
+              (and (current-local-map) (list (current-local-map)))
+              (mapcar
+               #'cdr
+               (seq-filter
+                (lambda (elem)
+                  (symbol-value (car elem)))
+                minor-mode-map-alist)))))
+        (or (command-completion-using-modes-p symbol buffer)
+            ;; Include commands that are bound in a keymap in the
+            ;; current buffer.
+            (and (where-is-internal symbol keymaps)
+                 ;; But not if they have a command predicate that
+                 ;; says that they shouldn't.  (This is the case
+                 ;; for `ignore' and `undefined' and similar
+                 ;; commands commonly found in keymaps.)
+                 (or (null (get symbol 'completion-predicate))
+                     (funcall (get symbol 'completion-predicate)
+                              symbol buffer)))
+            ;; Include customize-* commands (do we need a list of such
+            ;; "always available" commands? customizable?)
+            (string-match-p "customize-" (symbol-name symbol))))))
+
+
 (defun command-completion-button-p (category buffer)
   "Return non-nil if there's a button of CATEGORY at point in BUFFER."
   (with-current-buffer buffer
@@ -2502,7 +2555,11 @@ Also see `suggest-key-bindings'."
 (defun execute-extended-command (prefixarg &optional command-name typed)
   "Read a command name, then read the arguments and call the command.
 To pass a prefix argument to the command you are
-invoking, give a prefix argument to `execute-extended-command'."
+invoking, give a prefix argument to `execute-extended-command'.
+
+This command provides completion when reading the command name.
+Which completion candidates are shown can be controlled by
+customizing `read-extended-command-predicate'."
   (declare (interactive-only command-execute))
   ;; FIXME: Remember the actual text typed by the user before completion,
   ;; so that we don't later on suggest the same shortening.