I found that arbitrary terms in the RAW are more complicating than clarifying. What specifically are participants?  Are they the players?  The PCs? Monsters included? What about combatants?  Are they the same thing?  Why use a different term, if it means the same thing?  Plain simple language that is okay in casual conversation can generate a lot of second guessing in formal game rules.  Also, the terms the RAW uses seem to suggest a lot of assumptions that don't always hold -- i.e. sometimes the DM also runs NPCs on the PCs side of the battle. 

Here's my best attempt at something clearer to hand to new players at cons.  It has a few more lines, but the information is broken out into more digestible chunks:

The following terms are used in describing combat.  

Combatant -- every PC and NPC involved in a combat encounter.
Adventurer -- any combatant opposing the adversaries and run by a player. (Usually a PC, but players sometimes run NPCs.)
Adversary  -- any combatant opposing the adventurers and run by the DM. (There may be rare exceptions where an adversary is run by a player.) 

Ally -- any combatant opposing the adversaries and run by the DM -- i.e. an ally of the adventurers.
Initiative Group  -- any adversary or set of adversaries that makes one roll collectively for initiative. 

COMBAT STEP-BY-STEP

1. Determine surprise. The DM determines whether any combatant is surprised.
2. Establish positions. The DM decides where combatants are positioned relative to each other,
                                         where they are located as well as which direction they face.
3. Create initiative groups. The DM decides how many initiative groups there are and which adversaries belong to which group. 
                                                (Usually grouping is done by creature type.) 
4. Roll initiative. Adventurers, allies, and adversary initiative groups roll for initiative.  Roll results are compared and ordered, highest to lowest.
5. Take turns. Each combatant takes a turn in initiative order.
6. Begin the next round. When all combatants have had a turn, the round ends. Repeat step 5 until the fighting stops.

I can already see 1 potential problem. There are a lot of rules and features that refer to "allies", which will usually be PCs.

Sometimes terms are purposefully vague. The initiative system was made to be as flexible as possible so that it could be used in all kinds of situations. Trying to be more precise about language reduces that flexibility.

An encounter may start as a combat but transition into a social encounter or skill challenge. Using the term 'combatants' has the effect of pushing aside those methods of alternative resolution to the point that new players might not even consider it an option. Or you could have a situation with only PCs in the initiative order dealing with traps or hazards, or trying to escape or sneak around, or any number of other things. In those situations, 'combatants' wouldn't really be the correct term.

There are also situations your terms don't cover:

You have spells like conjure animals that effectively create PC-driven initiative groups as well as allies like the Battlesmith's Steel Defender that shares the PC's initiative and goes directly after them.

You could have participants that are completely neutral to the PCs and whoever is opposing them, or you could have a faction that opposes both the PCs and their adversaries.

You can have things like vehicles which are controllable by both sides under the right circumstances and have their own turns and actions and may or may not be 'combatants.'

You can have traps or environmental hazards that have no allegiance to anyone, they are just there and effect whoever is standing in the wrong place at the wrong time.

Yeah, the terminology used is kind of complicated. It needs to be in order for this system to attempt to handle literally anything our brains can come up with. I think it's worth it.

Ally is actually a term that is relative to the creature using the ability that uses the "ally" reference.

Quote from Farling >>

Ally is actually a term that is relative to the creature using the ability that uses the "ally" reference.

 Yup. Allies can include PCs, pets/companions/etc., NPCs, sidekicks, whatever.