Choice of Phono Cartridges

I have used the AT OC9III(discontinued, but still available) 
Not having compared specs in detail, it looks like the XML is similar, and with a threaded body, which the older versions lack.
Great cart for the money. There are plenty threads with satisfied users.

The Soundsmith Otello has a nicer mid/high presentation, but soft in the bass. Still a very nice cart in its price point. The SS web page points this deficit out. I use it as an emergency back up cart.

This naturally, is a subjective opinion-YMMV.

With that many albums, a more serious table setup and eventually, phonostage may be  something to consider.

Given the quality of your amps/speakers and size of record collection, I personally think you would get more enjoyment from your records by upgrading the whole turntable.

There are some great deals out there at the moment - for example Music Direct are specialling out the Rega RP8 for $1995 ( outgoing model, $1000 discount ) - brand new and ready to go. You can get a better front end second hand but why not avoid the hassle. Buy a Rega Exact and it is plug and play.

If you decide to upgrade the cartridge for now I would stick to MM/MI because they are less demanding of the tonearm ( which on the AT is not a world beater ). My recommendations in your price range would be

Soundsmith ( these are excellent at all price levels - very smooth, detailed and musical ).
Nagaoka 

I prefer these to both the Ortofon MM's and Sumiko's.
Whilst the AT MC's do perform well for the money, you wont get the best results with the AT arm.

+1 on dover's comment above

tt upgrade is best bet

try a well tempered labs or an oracle

Choosing a cartridge you must take in count an effective mass of your tonearm. Some cartridges will not work properly on your current tonearm because of the tonearm mass and cartridge compliance miss match.

Even if you can find inexpensive MC cartridge within your price range this is not final. You will have more problems with low output MC in a mediocre system. Another problem is a cost of factory re-tip (and you will have to ship your cartridge away for a long time).

Of course you can upgrade your turntable, but for every turntable you need a cartridge. You can easily start with a cartridge and if it’s a mid compliance cartridge then it is more or less universal for modern tonearms (they are mid mass).

In your situation I think MM or MI is a better choice within your price range. If you will add some NOS cartridges in your list you will get much more (such as exotic cantilevers for example) in the same price range. This is where some nice vintage MM are the best. I don’t know the mass of your tonearm (you have to check it) but a mid compliance MM cartridges from Japanese brand Victor in X1II series are superb and you will find a lot of great feedbacks from other users on audiogon about them!

Regarding new MM cartridge you can add Australian made Garrott Brothers p77i in the list.

I agree with others, stick with MM now so you can stick with mid or higher compliance cartridges for your existing arm, get a nice MM now, and later, when you move to a new TT, new arm, make sure the arm is good for mid/high compliance cartridge (likely).

sticking with MM, and Audio Technica

I have and love the sound of AT440ml microline stylus on aluminum shaft, currently using it every day on my office system.

AT540ml is the current version, microline/aluminum shaft/plastic body

IF you simply upgrade your existing cartridge’s stylus, it would be equivalent to 540ml

AT740ml is same as 540 except it is a cast aluminum body

https://www.audio-technica.com/cms/cartridges/09079f798b312248/index.html

the next logical step is a lighter/stiffer shaft like boron, ....
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don't forget the improved traceability of advanced stylus shapes, follow and stay in the grooves.