It is difficult to measure user engagement - you are right that sites like e-commerce sites don't measure changes in terms of usability or engagement but instead by a more easily measured metric (such as $$$).
You mention some good bits of data to collect (button clicks, time spent on a screen, etc.), but the key here is that you still need to define higher level engagement objectives, such as:
- How many users are registering for your application (or how many users are registering after using a demo version?)
- How many page hits are you getting on your application's call-to-action(s)?
- How much are users using Feature X in your application as opposed to Feature Y?
Once you have defined engagement objectives, you can then go about collecting the appropriate data (button clicks, page hits, timers, etc.) to answer your research questions.
If defining objectives sounds hard, that's because it is. Note that some people object to the idea of measuring user engagement because of this ambiguity.