Cogito - most of the "newer" Greist sets that are top clamp don't need a "special" adapter. By then, the machines came with it already mounted to the presser bar. These ones are sort of "in between" in age -
I don't know exact time frames here - it seems that pre 1900, US machines (other than Singer) mostly had feet that slipped on to the presser bar, and the hemmers and tuckers were bed mount. Then it came sets with an adapter that could be mounted to the presser bar, and the feet hooked into that - and usually the hemmer now mounted to the foot - and not the bed of the machine.
It wasn't long, and the top clamp seemed to become the method US (other than Singer) were using - and by this time the machines started coming with the adapter part of the machine, and even the standard foot was top clamp. Boye came out with a chart in the early 1900's that shows different top clamp types - as a way to help identify your machine so you would know what needle/shuttle it would take.
When US mfg shifted overseas, the top clamp appears to have become a thing of the past ( along with most of the companies that made them) and the Japanese made machines were standardized with Singer - except Kenmore - but then Sears still likes to be a bit "odd" so their stuff isn't quite like anyone else's.
The short of it - For US made machines (other than Singer) unless you have true antique machine, it's likely to already have the adapter and take top clamp attachments. Then your concern is they are machine specific - and if you have set that does not "fit" its for a different machine.
The ones that need adapters that came with the set are these odd ball sets - that are something other than top clamp feet. ALL of the second group of pictures are regular top clamp feet, just different machines or ages.