What does it mean to fly a procedure turn? So, I'm flying IFR to Newport (KONP) for the ILS RWY 16. There's an aircraft on the approach, so I'm cleared direct BUHRS at 5000' to enter the hold as published.

I get to BUHRS, I turn outbound on the hold, and am "cleared for the approach". (Note: Not "straight-in").

So, what is my next step?

AIM 5-4-9 says I'm required to fly the procedure turn, but I can't find a definition of what that actually means (i.e. do I have to overfly the waypoint to "start" the procedure turn, or does it just mean remain in the protected area?). The options seem to be:

1) Turn back to BUHRS, overfly BUHRS, turn right (ending up exactly back where I am right now), intercept the localizer outbound, descend to 2600.

2) Intercept the localizer outbound, descend to 2600.

If #2 is correct, then there's an additional point of consideration. 5-4-9a1 says I have complete discretion in how I fly my procedure turn. Certainly turning immediately inbound fits within that discretion (instead of turn right, intercept outbound, turn left, turn left, turn right, intercept inbound). Or, what about continuing to fly outbound by heading for a minute then turn inbound and intercept the localizer?

A simplistic reading of the AIM seems to indicate #1 is correct. I asked AOPA, they say #2 is correct. 

Is #2 correct? If it is, wouldn't an immediate turn inbound also be correct? What is the definition of "procedure turn"? Does it require flying over a fix to "start"? Or can you be anywhere in the protected area?

The easy solution is to ask for a straight-in clearance, but assume you're not in radio contact or your request is denied.
In my opinion, as soon as ATC clears you for the approach, the holding pattern at Buhrs is cancelled (i.e., is no longer required - 1 min. leg cancelled). Begin a descent to no lower than 2600  and turn inbound as soon as you are far enough north of Buhrs (remaining within 10 NM) that you can comfortably descend from no lower than 2600 to  no lower than 2100 after established inbound on the localizer. (I would plan to be reasonably close to 2600 at the point of intercepting the localizer inbound so that I would not find myself above the glide slope).  

Remain at or above 2100 until intercepting the glide slope and finish the approach.

As you have pointed out in your question, the "point," "type," and "rate" of turn is left up to the discretion of the pilot.  Since it is perfectly legal, and most importantly, safe to do a racetrack pattern (on the PT maneuvering side) and that is exactly the position you are in, this would be the best option.  

The fact that your approach clearance came shortly after you had entered the hold (when you were turning outbound) does not obligate you to go back to Buhrs and then officially begin the approach by turning outbound again (45 degree procedure, racetrack procedure, 80/260, etc.).

Just tell ATC you are leaving 5000 and beginning the approach.

From the AIM para. 5-4-9, sub para 1: