Thinking about it a bit more (and I’m very willing to be corrected), I think the main categories are more like:
1. Non-lead monolithic. Very hard, generally relatively low expansion, need higher velocity to expand, penetrate very well. Limited meat damage.
2. Bonded core lead. Hard. Often limited expansion, especially at low speed, but expand well at high speed or if hit bone. Penetrate well. Moderate to little meat damage, unless hit bone or at very short range.
3. Classic unbonded cup and core soft point. Quite soft. Usually expands well, except at very low speed or if misses bone in small bodied animals. Can have limited penetration at high speeds and/or hit bone, when can ‘explode’. Moderate to severe meat damage, depending on speed and whether bone hit on entry. Many of these have been recently 'tarted up' with the addition of a plastic tip, which is to improve accuracy rather than to do much to the terminal performance (things like Nosler Ballistic Tip fall into this category).
4. Modern long range/ very long range aggressively expanding. Designed to expand very fast and/or at very low speed. In calibres and bullet weights primarily designed for deer sized animals, these bullets are primarily designed to maximise accuracy and expansion at long range - so can be extremely aggressive at high speed/short range. Can penetrate well at very low speeds, but can also explode very dramatically at high speeds. Meat damage can be horrific at short range. Things like Hornady ELDM and ELDX fall into this category.
5. Varmint bullets. Usually very light-for-calibre to maximise speed. Designed to fragment/explode on impact with soft tissue. Primarily designed for use on small to very small bodied pest animals where the aim is to kill immediately, with no need to recover meat. Do not penetrate well at most speeds, though most will go through a roe deer, and many will go through a small hind if you miss bone. Meat damage can be spectacular, and render a small carcass unfit for consumption. These are the things that people probably mean when they talk about 'ballistic tips' and head shooting. Things like 110gr VMAX in .270 or 75gr VMAX in .243 fall in this category - though even here, at longer range or if downloaded to slower speeds, they behave more like the category above. Because they're often crazy fast, they were favoured by people doing a lot of culling because they reduced the need to worry about range estimation. I get the sense that they are slowly dropping out of favour as people use range finders more and dial in.