Use of the word "thin" for pads is confusing. What was marketed as "standard" pads (around .185" = 4.7 mm) for decades were actually too thick for many saxes, especially modern saxes, unless you put in a thick bed of glue and tilted the pad in the cup, something that most technicians with standards were rather reluctant to do. These pads tended to be quite squishy, so I suppose heavy pad impressions would help to get them to seal, but heavy impressions tend to be unstable. People lived with that. These deep impressions would have reduced their effective thickness by up to around 1 mm.
"Thin" pads were about 0.165" (= 4.2 mm), and purchase and use of that thickness was a lot less common.
Expectations of pads seem to have advanced a long way. We now expect the "feel" of firmer pads sealing well with a light touch, which implies having very consistent felt and leather thickness, level tone holes, very little "give" in silencing materials of linkages, secure post mounting, and accurate pivots.
So firmer pads are now the norm, and the norm for thickness would be around 4 to 4.2 mm. Impressions tend to be very shallow. This thickness is now suitable for most saxes, so they are regarded as "standard" rather than "thin", because they are in "standard" use.
So now I expect any pad supplier to specify the actual thickness rather than using vague terms like "medium", "standard", or "thin". I certainly would not buy from a supplier who did not specify thickness.
Much the same thing has happened for flute pads. (Along with the problem of non-level rolled tone holes being incompatible with firm pads if you want reliable adjustment. The norm is for rolled tone holes to be non-level.)