Markus, thanks for another detailed suggestion 
Considering you use the Track browser extension most of the time, the options here are a bit limited in terms of features, how many past entries are visible in the extension pop-up, and how autocomplete search works.
If you would consider using the web app, which you can have as a pinned tab in the browser, for example, you can use the Favorites feature to pin a timer as a favorite in the app, at the top of the Timer page, and just click on that every time you want to track time with the same exact details.
If we are focusing only on the extension at the moment, your suggestion will be noted as a feature request and shared with the team. I hope we can meet your needs better in the future. Thanks for helping us with suggestions, and keep them coming! 