FWIW, I have this in my init file since forever:

(setq my/set-cursor-color-color "")
(setq my/set-cursor-color-buffer "")
(defun my/set-cursor-color-according-to-mode ()
"Change cursor color according to some minor modes."
;; set-cursor-color is somewhat costly, so we only call it when needed:
(let ((color
(if buffer-read-only "#8888FF"
(if overwrite-mode "#000000"
"#FF0000"))))
(unless (and
(string= color my/set-cursor-color-color)
(string= (buffer-name) my/set-cursor-color-buffer))
(set-cursor-color (setq my/set-cursor-color-color color))
(setq my/set-cursor-color-buffer (buffer-name)))))
(add-hook 'post-command-hook 'my/set-cursor-color-according-to-mode)

Cool. Will it be possible to replace these settings in my init file
with the new package?

#+begin_src emacs-lisp
(defun my-change-cursor (&rest _)
"Change cursor color and type depending on insertion mode and input method."
(set-cursor-color
(cond (repeat-in-progress "blue")
(current-input-method "red3")
(t "black")))
(setq-default cursor-type
(cond (overwrite-mode 'box)
(t 'bar))))
(add-hook 'post-command-hook 'my-change-cursor t)

;; Also update the cursor on the repeat timer events:
(add-function :after repeat-echo-function #'my-change-cursor)
#+end_src