Reddit mentions of Ortofon Super OM 10 Moving Magnet Cartridge

I'd recommend the Ortofon Super OM 10, it's the one I personally use and it falls into the upper end of your budget at $72.

https://www.amazon.com/gp/product/B000CCEQM4/ref=oh_aui_detailpage_o03_s00?ie=UTF8&psc=1

Or the previous version Ortofon Super OM10 for $20 less and is basically the same except for the red plastic.

The better upgrade is to the Nagaoka MP-110 $129.

Why not spend the $50-$100 its going to take to put a headshell and decent budget cartridge on the Gemini? That's cheaper than a night out, and you'll get a lot of hours of enjoyment out if it (setting the turntable up, tinkering/learning, and playing records). Even if you upgrade later, it's better than letting the Gemini collect dust or throwing it in the dumpster and raising the amount of perfectly serviceable and fun e-waste rotting in our landfills.

Edit: You will need to fix the speed adjustment slider and buttons on at least one of those though. That's very important for a record player.

Audiophiles are very picky and tend to invest a lot in their equipment (especially if they're buying brand new high end manual turntables, which is what this subreddit has a fetish for). Yeah the Gemini and the Sony look like shit if you sit them next to a brand new Rega, but why should you care?

Your Sony is cheap to buy, very simple and cheaply made, therefore to an audiophile it must be bad. It is also nearly impossible to tinker with (which is probably what is causing your dissatisfaction). That being said, it plays records and plays them pretty well. Your new turntable(s) won't give you much improvement in audio quality, but it will give you a ton of room to have fun with the hobby.

The Gemini is what is referred to as a Technics SL-1200 clone or Super OEM, and your assessment of it is mostly correct. It looks almost like an SL-1200, and it functions almost like one, but it has NONE of the refinement. Nice price though. It's actually pretty comparable to an Audio Technica AT LP-120, which costs a lot more than $15. Like I said, the Gemini probably won't give you much in terms of an audio improvement over the Sony. It will require a phono preamp if you don't have one yet (the Sony has a built-in preamp, the Gemini does not).

Changing headshells is a huge part of the fun of vinyl, and definitely something you want to get into if you're wanting to get technical with the hobby. The AT95e is one of the best budget cartridges and the Ortofon OM10 is the other. Since you specifically said you want to tinker, I would not buy the preassembled cartridge. I would buy a seperate headshell and an alignment protractor and do the installation myself. It's slightly cheaper than spending the $60 for the ready to install kit, and you'll learn more that way. If you're intimidated, there are A LOT of youtube guides on how to do this.

The Ortofon 2M Red and Shure M97xe are slightly classier budget cartridges, but they won't offer a big performance boost over the good budget option (the Ortofon Red sure is pretty looking though...). You'll need to spend >$200 on a cartridge before you start to see a significant upgrade over the budget options.

edit:
Quick links -

Budget headshell - https://www.amazon.com/ADJ-Products-TT-HEADSHELL-Turntable-Cartridge/dp/B0002E51V2

Alignment protractor - https://www.amazon.com/Turntable-Phonograph-Cartridge-Alignment-Protractor/dp/B079ZBLJ4M

AT95e - https://www.amazon.com/Audio-Technica-At95e-Phonograph-Cartridge/dp/B004NRVUMI/

Ortofon OM10 - https://www.amazon.com/Ortofon-Super-OM10-Phono-Cartridge/dp/B000CCEQM4/

Ortofon 2M Red - https://www.amazon.com/Ortofon-2M-Red-Phono-Cartridge/dp/B000WMCEKK/

Shure M97xe - https://www.amazon.com/Shure-M97xE-Performance-Magnetic-Cartridge/dp/B007ZC2EYQ

AT LP-120 - https://www.amazon.com/Audio-Technica-AT-LP120-USB-Direct-Drive-Professional-Turntable/dp/B002S1CJ2Q/

Here's my recommendation, because it's upgradeable to the sky by just buying a more expensive stylus in the OM line: https://www.amazon.com/Ortofon-Super-OM10-Phono-Cartridge/dp/B000CCEQM4

I'm a huge fan of Ortofon. This one is great value for money.

https://www.amazon.com/Ortofon-Super-OM10-Phono-Cartridge/dp/B000CCEQM4

• http://hifiheaven.net/shop/Shure-M92E-Value-Phono-Cartridge?language=en&currency=USD&gclid=Cj0KCQiAoJrfBRC0ARIsANqkS_4VrMvAOkj_eotXlcBDB45s7Es9S8aC2Tqb7b2VEERuH6hlig1CPlQaAsFLEALw_wcB
  
  
• https://www.amazon.com/Ortofon-Super-OM10-Phono-Cartridge/dp/B000CCEQM4