It would be nice to allow customizing how mode-line-buffer-identification is generated, since it's quite long. If the file has been moved it gets even longer, since the "name: " will then also show the path.
For example, I would like to shorten the "timemachine:" prefix and remove the date from the end so that the mode line shows something like "T:foo @hash name:foo (1/2)" instead of "timemachine:foo @hash name:foo (1/2 2 years, 5 months ago)". For my use case, adding git-timemachine-mode-line-prefix (string) and git-timemachine-mode-line-date (boolean) custom variables would work.