Machines sold by Sears get quite interesting LOL!!
Early on, Sears used the name Minnesota as the badge, and over time had several different Mfg make machines with that name on them.
As to top clamp attachments, they are not interchangeable, and the differences are often subtle, but you can use things like the shirr plate, or included bobbins to help guess what they fit.
White as the wider slot, center needle hole and a slightly longer (think it's 5/8") distance between gap and needle hole. So far, White is the only one I've encountered that has a centered needle with the longer distance.