Thinking in terms of World of Warcraft, the next incentive after building up a reasonable set of rewards is to upgrade, getting something bigger and shinier to replace something you already have.

A right-click menu option would solve my needs.

Looking at the 4 other Chrome extensions I have installed at the moment, 2/4 have right-click menu options and 4/4 have a dropdown on left-click. Either one would be a solution for me, as long as I can start a timer without looking at my city.

According to the Pomodoro technique (which I'm assuming you are imitating), you don't get any partial credit for any incomplete focus sessions, you just have to start a new focus session.

The last thing we would want in a productivity app is an interruption to ask "Are you still there?". Perhaps an alternative would be just a question in the notification at the end of the session to ask "Were you focused, or were you distracted, or were you interrupted?"

I do rely on the notification popup to start a new timer or take a break, so that workflow works well.

Out of your 3 options, I like the second one the best. The popup could be as simple as those two buttons, or it could be similar to the existing notification popup.

The main thing that I would like to be able to do is start a timer without displaying my city to the office. I can appreciate the benefit of gamification in the workplace, but I'm not totally sure my boss would be on board.