The Brill Dictionary has the following classification of the aorist (the examples are mine): sigmatic - ἐποίησα, thematic - ἔλιπον, and root - ἔβην. Are my examples correct?
I guess what he calls the thematic ao is usually called the second ao in Sm and elsewhere.
Yes, those examples are correct. Sometimes “root aorist” will be called “athematic aorist” or “athematic root aorist.” They all refer to the same thing. The label “2nd aorist” conflates root and thematic, which is why it’s generally avoided by more recent scholarship informed by linguistics.