I use
KanbanFlow for project planning and time tracking. There are lots of features for managing tasks, planning, estimating, whatever. But the beauty of a tool like this is it has enough base features that you can customize for your use. For example, just for time tracking, you could have 4 or 5 items in your list, select one, start the timer.
I don't track for anyone buy myself, and just use it to get some general insights. I have 3 monitors, and mostly dedicate one to project tracking, and I find I rarely forget to start the timer. If this data will be audited by others, maybe a more strict tool would be best. I'm thinking something that locks down your computer unless you are actively billing towards a project
A similar and probably more popular tool is
Trello. Trello doesn't have time tracking built-in, though there are some free browser plugins that add it. Both KanbanFlow and Trello are free with many features, and have some paid options for more features (mostly around multiple users and reporting).
For productivity, try the
Pomodoro technique. The basics is to commit to a set period of focus (suggestion is 25 minutes), then take a full on break (suggestion is 5 minutes). I find committing to 25 minutes of focus at a time is easier than thinking "1:00 PM. OK, time to doing something towards that 2 week task...". Similar to your tool, some also like the idea of a physical timer for the Pomodoro technique. There are many apps, programs, web apps, etc.
Finally, for general productivity, I've found that Microsoft OneNote is the best for notes. It has clients for Mac, Windows, Android, iOS, but unfortunately not Linux. And it syncs all your notes to the free Microsoft cloud.
EDIT
Hmm, it seems one of the premium features of KanbanFlow is a Time spent report. If you primarily want this tool for time tracking and reporting, the basic time tracking in the free version might not be suitable for you. Well, there is a free trial for the premium service