What exactly distinguishes a beacon light from a strobe light? Different sources seem to have different opinions:

Wikipedia: Aircraft warning lights:


  These lights include landing lights, red or white flashing beacons, wingtip strobes, and wingtip navigation lights.



Beacons and strobes are flashing lights.
Beacons may be red or white.
Strobes are wingtip lights. I interpret this as strobes being directional while beacons cover 180°.


Aeronautical Information Manual (AIM), 4−3−23. Use of Aircraft Lights:


  b. An aircraft anti−collision light system can use
  one or more rotating beacons and/or strobe lights, be
  colored either red or white, and have different (higher
  than minimum) intensities when compared to other
  aircraft. Many aircraft have both a rotating beacon
  and a strobe light system.
  
  d. Prop and jet blast forces generated by large
  aircraft have overturned or damaged several smaller
  aircraft taxiing behind them. To avoid similar results,
  and in the interest of preventing upsets and injuries to
  ground personnel from such forces, the FAA
  recommends that air carriers and commercial
  operators turn on their rotating beacons anytime their
  aircraft engines are in operation.



Beacons are rotating lights. They're always referred as rotating beacons in the AIM.
Both beacons and strobes can be either red or white.
I take the sentence Many aircraft have both a rotating beacon
and a strobe light system. as beacons being the lower intensity lights which are constantly turned on during engine operation signalling the danger of an aircraft in operation, while the strobes being the higher intensity flashing ligths with the reason of being seen from great distance.


Wikipedia: Aviation navigation lights


  Aircraft navigation lights are placed in a way similar to that of marine vessels, with a red navigation light located on the left wingtip leading edge and a green light on the right wingtip leading edge. A white navigation light is as far aft as possible on the tail or each wing tip.[3] High-intensity strobe lights are located on the aircraft to aid in collision avoidance.[4]
  
  In civil aviation, pilots must keep navigation lights on from sunset to sunrise. High-intensity white strobe lights are part of the anti-collision light system, as well as the red rotating beacon.
  
  All aircraft built after 11 March 1996 must have an anti-collision light system (strobe lights or rotating beacon) turned on for all flight activities in poor visibility. The anti-collision system is recommended in good visibility, where only strobes and beacon are required. For example, just before pushback, the pilot must keep the beacon lights on to notify ground crews that the engines are about to start. These beacon lights stay on for the duration of the flight. While taxiing, the taxi lights are on. When coming onto the runway, the taxi lights go off and the landing lights and strobes go on. When passing 10,000 feet, the landing lights are no longer required, and the pilot can elect to turn them off. The same cycle in reverse order applies when landing.



Beacons are red rotating lights.
Strobes are high-intensity white lights.
Beacons stay on for the duration of the flight including the whole engine run time.
Strobes are added from takeoff until leaving the runway after landing.


Is there a clear definition?
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.