log message button state when new messages

[TurboGit]

It would be nice if the "list button" could be enlightened if there are new logs since last opening.

Originally posted by @jenshannoschwalm in #21171 (comment)

Remains only the code clean-up (see below).

[TurboGit]

Also to be done: clean-up the code as we don't need anymore to display more than 1 line of log message.