I totally agree with the 3 choices you mentioned, but I would add 4 more.
Choice #4 - High speed or slow.
Choice #5 - Extremely precise details you get with high quality print output or accepting less precise, lower quality output. (This may be the same as your #2, but I wasn't sure.)
Choice #7 - Size of the print volume. What is the maximum sized object your printer can print in one piece?
Choice #8 - Level of customer support you are willing to endure/accept.
If you want high speed, it generally costs a bit more and can sacrifice print quality. If you're ok with medium to low speed, you can get higher print quality (potentially.) If you want both speed and print quality, you will pay more than the price of your car to get it (think Carbon3D or NewPro3D). If you want a company with great customer support, you will also pay extra for it, in most cases.
Build volume size also changes the final price you can expect to pay. My current printer can print a 305x305x305mm cube. (1 cu.ft.) I'm always on the lookout for another printer with even bigger build size, with similar reliability, print quality, etc. I'm worried at the same time, that I might have to trade something else I need (print quality? speed?) to get a bigger build size for reasonable coinage.
I've owned 4 printers ranging from a $2k build it yourself kit, up to a plug-n-print professional level machine at $4.5k. Can you guess which one I'm still using after donating the others to a local maker space? Yeah, it's the pricey one. I kick myself thinking, "if only I had bought that printer first, I would have saved about $10 grand spent during the 4 year learning process." The pricey one requires far less fiddling to keep it running. I run 24-48 hour prints on it regularly, without fails. The company still honors the warranty even if you have modified the machine. They sell an add-on device which let's you do multiple color printing using a single nozzle. I even run the printer faster than normal without problems, after putting tighter, stronger belts on it. I spend far more time designing and printing rather than working on this printer. That said, there is ALWAYS regular maintenance to be done on any mechanical machine, so you WILL need to do some fiddling with just about any printer.
Despite all of these seemingly scary things to keep in mind, there are some really fine printers out there for a moderate amount of money these days. There are more quality choices now, than ever before. Personally, I would steer around most of the under $800 printers, if you can. You can get something pretty exceptional for $1000-1500 these days.