What is meant by "Outbound leg timing begins over/abeam the fix, whichever occurs later" in FAR/AIM 5-3-8?
The text is written that way to take into account the various type of holding entries. In case of a parallel or teardrop entry, the outbound leg starts after passing over the fix. In all other cases the outbound leg start after passing abeam the fix, following a turn that starts over the fix. 

Case 1: Established in a holding pattern

So when established in a holding pattern, the outbound leg starts abeam the fix.

The sequence is:


inbound leg
pass over the fix
turn outbound
pass abeam the fix
outbound leg
turn inbound
repeat from 1.


In this case it is clear that the time of the outbound leg should start when you pass abeam the fix, which occurs later in the sequence. 

Case 2: Entry into holding pattern



Case 2.1: Parallel Entry into holding pattern

When entering the holding using a parallel entry ((a) in the diagram) the sequence  is:


fly to the fix
pass over the fix
turn outbound course
outbound leg (on non-holding side)
turn to inbound leg
intercept inbound leg
follow sequence for established in a holding pattern


In this case the timing of the outbound leg should start when you pass over the fix. From this, it follows that the outbound time will include the time spent in the outbound turn (3.). 

Case 2.2: Teardrop Entry into holding pattern

When entering the holding using a teardrop entry ((b) in the diagram) the sequence  is:


fly to the fix
pass over the fix
turn outbound course
outbound leg (30 degree into holding side) 
turn to inbound leg
intercept inbound leg
follow sequence for established in a holding pattern


In this case the timing of the outbound leg should start when you pass over the fix. From this, it follows that the outbound time will include the time spent in the outbound turn (3.). 

Case 2.3: Direct Entry into holding pattern from holding side

When entering the holding using a direct entry from the holding side ((c) in the diagram, sector on the holding side) the sequence  is:


fly to the fix 
pass over the fix
turn outbound
pass abeam the fix
outbound leg
turn inbound
follow sequence for established in a holding pattern


In this case it is clear that the time of the outbound leg should start when you pass abeam the fix, which is later in the sequence.

Case 2.4: Direct Entry into holding pattern from non-holding side

When entering the holding using a direct entry from the non-holding side ((c) in the diagram, sector on the non-holding side), you may not complete the turn before being abeam the fix. In that case the sequence is:


fly to the fix 
pass over the fix
turn outbound course
outbound leg
turn inbound
follow sequence for established in a holding pattern


In this case it is clear that the time of the outbound leg should start when you pass over the fix. From this, it follows that the outbound time will include the time spent in the outbound turn (3.).