LF, MF, and so on are bands, or frequency ranges. AM is a modulation technique. It, like FM and others, can be used on any band (although regulations can limit this).
It doesn't: There's a lot of non-AM/non-voice stuff on longwave, most notably various digital time beacons (WWVB in the US, DCF77 in Western Europe etc.)
The US often just says 'AM radio' to refer to the MW band, because they don't have commercial broadcasting on the LW band, so there's no need to disambiguate. In the UK we usually specify LW or MW because both bands are in use for 'AM radio' (for now ..).
Same way we just say FM for the 88-108 broadcast band. Were there two FM broadcast bands, we'd need to give them different names.
Well, yes and no. Inside specific areas of all bands there is a convention to use specific modulation. On the Amateur bands, whilst you are free to use whatever you like, there are conventions that CW, digital, SSTV, SSB and FM are generally grouped into sub- bands. It helps people find each other, so is pretty useful. If your receiver is set to FM you might not notice a CW signal. On other commercial bands the mode is enforced.
