Admittedly lengthy but useful to be aware of, here is some insight about MIDI and MIDI files. (No need to know everything by heart)
By type of MIDI file, I (mostly) meant what data it was actually containing. Those data depends a lot on the way the MIDI file was created (and its creator’s purpose/proficiency). Therefore, depending of the way the MIDI file was created, it may not contain all the data that Logic will need to render it satisfactorily (to fulfill your specific needs). Which might (in your case) imply some adjustment(s) to achieve interesting score layout. The required adjustment(s) may range from little to extensive, sometimes none. AFAIK, that also applies to all DAWs.
For the source of MIDI file types, googling "MIDI files", with descriptors to narrow the results will yield a slew of options. Some are free, others not. Quality varies a lot. I personally don’t have specific website to recommend. This website seems however promising…