What exactly distinguishes a beacon light from a strobe light?
I think there are two points in your question: what is the physical difference between a beacon and a strobe, and what is the operational difference? For the rest of this answer, I'm going to assume that you're asking about an aircraft that has both a beacon and strobes; as you quoted above, it's possible for an aircraft to have neither. I'm also going to assume you're asking for an answer that applies in the US; regulations in other countries may be different.

First, the physical difference is exactly what you've already determined: a beacon is a (relatively) low-intensity rotating red or red/white light; a strobe is a high-intensity flashing white light. (As noted in the comments below, some beacons flash instead of rotating, but they're still lower intensity than a strobe.) This isn't formally defined anywhere (e.g. 14 CFR 1.1 doesn't define them) but the information you found in the AIM is exactly how everyone I know uses the terms. It's also how you find those words used on aircraft part sites, e.g. beacons and strobes. So whatever the 'strict' definition might be, the practical and accepted one is exactly what you've found out.

Second, the operational difference essentially comes down to the question "when do I have to turn them on?" and in addition to what you found in the AIM the FAA gave more details as part of this legal interpretation. The questioner asked two questions about beacons and strobes: when to turn them on, and if they both have to be turned on. The key regulation quoted is 14 CFR 91.209, and note that the FAA considers the beacon and the strobes together to form a single "anticollision system":


  [...] it appears that the strobe light and the rotating beacon are
  part of the same anticollision system.


When to turn them on:


  91.209(b) requires that an aircraft's anticollision lights be turned
  on once that aircraft's engine is started for the purpose of air
  navigation. As a safety precaution, the anticollision lights should be
  on before starting an engine or causing a propeller or rotor to move.


If both have to be turned on:


  [...] because section 91.209(b) does not contain an exception for
  alternative sources of anticollision lighting, turning on the
  anticollision beacon would not relieve a pilot from the requirement to
  turn on the anticollision strobe lights. See § 91.209(b). However, §
  91.209(b), does give the PIC the discretion to turn off the anticollision beacon and/or the anticollision strobe light system if
  the PIC determines that it is in the interests of safety to turn off
  either one or both of these components of the same lighting system.


In other words, the regulations say you should turn them both on, but as PIC it's up to you to determine if using them is safe or not, and if not then you are allowed to turn (or leave) them off.

Practically speaking, I've always been taught to turn the beacon on before starting the engine because it's there to tell people near the aircraft that the engine is running or about to be running. The strobes should go on any time you need to be highly visible (which is most of the time), but not if using them would blind someone else or even yourself, e.g. taxiing or flying in clouds. The AIM 4-3-23 has more details on that, in addition to the parts that you quoted.