Mec-Gar makes great mags, and even produces them for some of the big firearm manufacturers anyway. If Mec-Gar says they for the MKIII, then they will work 100%. I have Mec-Gars for my CZ75 ad they are top notch.
I can't speak for specific differences between MKII and MKIII, but I tried a Mec-Gar on advice here for a second magazine. I don't have a bunch of time at the range yet, but they appear to be equal performers.
That said, I just ordered a 3rd magazine from ePay, a Ruger part. Cost was less than $25 shipped, and I see MK III pieces in that same range. Mec-Gar parts are at or less than $20, and I think it would be hard to go wrong with either. If you have any doubt, an extra $5 will fix it.
I have 10 older MK III magazines that are superb. I just bought a MK III Hunter that came with 2 Ruger mags and they are terrible. The spring sticks in each mag when loaded with 10 rounds plus they feel sort of gritty. I'm going tear them down see if I can improve them but when I need replacements I'll probably go with Mec-Gar.
The Mec-Gar mags I bought needed the feed lips very slightly opened up. They would not seat with the bolt closed. Other than that they performed as they should. Found them for $17 each so I was willing to take a chance.
May our lives glorify God,
I have purchased my extra 22/45 mags here. The Meg-Gars have never failed me. Greg Cote, LLC
Also look into the McFadden clip loader for the Ruger 22 mags.
The best $20 I have spent on my shooting hobby, it is such a time saver.
There are YouTube videos about using this.
I have the two original oems that came with my 22/45 and 2 MecGar from greg Cote. Except for the smaller slider button on the MecGars, I have found no difference. All 4 of mine I have done the mag tune up on (technique found elsewhere on this site) . I have a total of 3500 rounds through all 4 mags (just short of 1000 through each), no FTF from any of them. They get cleaned after 250 rounds each. There was a big difference in price when I got mine but there certainly has been no difference in performance.