I've recently added error tracking to my self-hosted analytics app (UXWizz), and the way I did it is simply add extra events to each user/session. Once you have the concept of a session or user, you can simply attach errors or logs as Events stored for that user. This solves the main problem mentioned in the article, where you don't know what happened, plus being an Event stored in a MySQL database, you can still query it.

Why not simply use Events for logging, instead of plain strings?