Solved!
The Homelink firmware in the 2008 Lexus system is not completely compatible with Chamberlain's Security2+ remote transmitters.
This issue has been described elsewhere. Chamberlain updated their security firmware in 2011 (I think) and Homelink systems installed prior to that update are problematic.
I ordered the remote transmitter shown in the GDO User Manual ... LiftMaster 373LM ... and it programmed the GDOs, the Lexus Homelink buttons, and the rolling codes as described in the Lexus User Manual and GDO/Homelink websites correctly.