Just to add to the answers already given; there used to be a practice to call "CQ 40m" indicating you are indeed on the 40m band, avoiding stations to comeback to your call if your transmitter would have harmonics... say in the 15m band...
This was obviously a long time ago, when filtering harmonics was more difficult, and more difficult to measure/tune when building/operating equipment... and probably license conditions were different then today's.
Still this practice can be heard on the bands today, although modern transmitters would not have harmonics (or should not have harmonics) outside the band. I guess old habits die hard, especially in Amateur Radio.
I guess the practice referring to the bands is also a matter of "least effort to indicate". Which is already discussed in previous answers. Far more practical to say "I had some contacts on the 20m band" is stead of "I had some contacts between 14.000 MHz and 14.350 MHz"
or band name (ie VHF, UHF) to describe the signal.
To answer that part of the OP question: the indication of VHF/UHF, and respecively HF would be a too broad indication to be accurately describing the operating band.
If you would take HF alone, it would be 9 (or 10, or 11 bands) which are included, namely: 80m, 60m, 40m, 30m, 20m, 17m, 15m, 12m, 10m. Some will count the 6m band as HF and some will count the 160m as HF. Although I believe 6m is actually inside VHF and 160m is actually MF.
VHF would have 3, or 4 bands; 6m, 4m, 2m, and some regions will have an allocation at 220MHz.
And so on so forth; so the indicators "VLF, LF, MF, VHF, UHV, and higher" are not suitable for use for Amateur radio to indicate where you are operating, they do have their uses, and are used if a "broad term" is sufficient in the context of what is communicated...