Just skimmed through the other thread. I think you are already almost doing the correct thing. Keep in mind:
So in your model above the residual correlations would be captured by doing something like y ~ 0 + event + (1 + event | subject) adding an intercept to the subject level. this would give you a varying intercept per subject, which is common across event types (if this number is higher, you predictions for both events types is higher, if it is lower your predictions for both event types is lower → residual correlation). If you think about the “random/mixed effect” as composite error, this might become more obvious. There is a cool chapter in Data Analysis Using Regression and Multilevel/Hierarchical Models (Gelman/Hill) – I think it is chapter 12.5 – that further describes this “view” of hierarchical models.
Hope this helps!
Cheers,
Max