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The PSR-e423 has some features that only seems to work with some MIDI files, and not just the random ones I download from the internet.
Some formats have the ability to show the notes on the LCD display, which in turn, teach me the song, and the song's tempo.
I'm trying to find a search term, conversion tool, or some magic word that lets me identify compatible MIDIs and non-compatible MIDIs
While I can't offer any information on your specific model of Yamaha keyboard, here are some resources you can use to learn more about Standard MIDI files in general. I think your question has more to do with that anyway.
Interest in Standard MIDI files is not what it used to be. Here are some resources.
Wikipedia article explaining the difference between Format 0 and Format 1 Standard MIDI Files.
MIDI File Format Specification from the MIDI Manufacturers Association.
MIDI File Format explanation from The Sonic Spot.
The Yamaha Musicsoft web site--one of the last companies in the business of selling commercial Standard MIDI files in various formats, usually for use on their digital pianos.