Capn Bloggs:
My understanding is that GLS (the "GBAS Landing System") uses normal GPS from the satellites and is augmented only by ground-based gear eg a/the LAAS, as you say, close to the airport.
You are correct. But, the original concept included WAAS, more or less.
From the FAA site:
Satellite Navigation - NAS Implementation
NAS Implementation - Procedures - Ground Based Augmentation System (GBAS) Landing System (GLS)
"GLS" is the acronym for Ground Based Augmentation System (GBAS) Landing System.
GLS was originally published as a placeholder for both Wide Area Augmentation System (WAAS) and GBAS minima and marked as N/A since no minima was published.
As the concepts for GBAS and WAAS procedure publication have evolved, it was decided that GLS will now only be associated with GBAS and will be on a separate approach chart. For more information on GLS approach procedures, please refer to the AIM section 1-1-21 and section 5-4-5.