For the most part, the multiple trade names for the same company arise when one company buys out or merges with another. They don't want to rebrand 500 dealers on the east coast, so they just keep them humming.
Thomasville is the name that Home Depot licensed (from Thomasville furniture) to put on Masterbrand cabinets (AKA Kemper, Diamond, Schrock, Thomasville).
All the same manufacturing plant, different trade names. HOWEVER the different trade names are not equal. Home Depot "dumbs down" the construction. Thinner cabinet side walls, charging 20% for the plywood upgrade then throwing it in for "free", sneaky things like that. Each of the lines has a small handful of door shapes that are exclusive to that line. The same door shape has different door names between lines. Point is, don't think that because the Thomasville build construction looks flimsy, that a Diamond kitchen will be the same.
Medallion is Schuler. Same company, same plants, same finishes.
Between the two, Medallion/Schuler (owned by Elkay) has different levels of construction. The lower "Silverline" is what I would consider a higher end "builder grade". Limited cabinet sizes and fewer modification options. Keep it simple-stupid. But keep the bang for buck factor high.
When you step up to Schuler Designer / Designer gold, that's where they really start to pull away from Thomasville. Better finish quality, and the ability to do cabinet customization. Where, if you needed something out of the ordinary for lets say a boot bench, or an entertainment center, with Diamond you might have to buy a bunch of parts and make modifications on-site. With Schuler, for the most part, if you can draw it, they will make it. Yes it costs a little more for your materials that way, but you'll have zero waste, it will be finished properly, and you won't have to screw around with a thing when it gets to your project.
Bottom line, you get what you pay for, and educate yourself. An entry level Toyota has a nice paint job but the car's not made to last. A jeep looks ugly to some people but you could roll it off a cliff and it'd still drive away. Be clear to yourself about what is most important to you.
Hope this helps.
From a cabinet dealer 18+ years, with currently more than a dozen manufacturers including these two above.