Number 2 is standard. In my experience It's a form of logging more than a form of data collection.
- Data collection is about learning things about your users - Logging is about learning what your users were doing when Something Went Wrong.
Obviously, there's a _ton_ of overlap here, in that a lot of (all of?) the info one uses collects for one purpose could be used for the other. As a developer of the UI, however, my main concern is not about learning more about our users (our UX people already do that), but rather about understanding what happened, or which features are actually being used (so that we can know if they are safe to prune).
If a company is hitting an external API with every interaction, I'm guessing it's for gathering data on user behavior. The question then is whether or not it's anonymous.
Note, most sites that do analytics don't really care about who you are, individually, as a person, just how can they categorize you and your activities so they can generalize it and find ways to "improve the user experience" (read: improve how many users spend mucho dinero on our services).
And it's gotten to the point that they're never satisfied and want as much data as they can get away with so they can analyze it from a billion angles and tell the developers to change everything about the app every other week in the chase for better numbers.