There are 2 definitions of "round" in 5e. There is the definition of "round" that you quote, which is from initiative position intMax to initiative position 0 (or negative if you manage that some how).
Then there is the definition used here for readied actions. This is the same definition used in "once per round" effects such as certain powers. This definition begins at the beginning of your turn and ends at the beginning of your next turn.
Because (unlike in 4e), readying an action and activating it, doesn't move your initiative order position, and because it matches the readied action refresh timer (which happens to use the same 1/round definition of round), readied actions can be used at any point in either the current round or the next round prior to your turn.
Here's how reactions refresh:
When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction. (Player's Basic p70)
This is a large part of the support that leads me (and many others) to believe that readied actions can roll up to your next turn.
You were right!
In that type of situation, everyone is ready to begin fighting at any moment. Everyone basically is 'readying' for their fight - not just the PCs.
Readied Actions
These actions do require a trigger to be specifically stated, as well as the action they will take if triggered. In a general situation, most Readied Actions take place during combat when the turn structure is in place. However, it may be possible to ready an action outside of combat, but you as the DM will need to adjudicate the possibility of that.
From the Players Handbook, page 193 (emphasis mine)
...you can take the Ready action on your turn so that you can act later in the round using your reaction.
First, you decide what perceivable circumstance will trigger your reaction. Then you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it...When the tirgger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
Your situation
In order for the players in your situation to their ready action, they would need to have explicitly stated what their trigger was and what the use of their reaction would be. However, given that the combatants were also likely 'readying actions', the initiative roll works because everyone is on even ground.
Even if this was a one-sided readied action event (where the combatants whom the PCs were fighting were not prepared for a fight), the trigger for the PCs readied action is the attack - which needs to be completed before they can act.
In addition, a readied action is just that, something that is being held waiting for the trigger. That could be a caster having started their spell and waiting for the event to finish casting it (if no event, the spell slot is still lost), or the bowman with their arrow nocked and bowstring pulled and ready to fire, etc. etc. That sort of action may also move towards a specific resolution (FIGHT!) rather than hoping for an alternate resolution (NEGOTIATE!)
The developers have your back
Jeremy Crawford Tweeted about when Readied Actions are available.
The options, including Ready, in the "Actions in Combat" section (PH, 192–93) are meant to be used in combat, after rolling initiative.
but he also suggested an alternative:
Your readiness can guard against being surprised. Otherwise, you roll initiative as normal. The DM might give advantage
Jeremy also had a nice discussion on this podcast about readied action and initiative (starts at about 6:10).
Unspecified Initiative Rolling
There is a potential issue in how you roll initiative that should be mentioned. This section is not a judgement against the actions described, but merely a reminder to have a consistent and agreed upon method of initiative at your table.
If you are trying to create a level playing field, then you should roll specifically and openly for each combatant. While the DM does have "control" to make changes behind the scenes, what you have done is to allow the situation to play out as you wanted it to rather than be dictated by the rolls of the dice.
Even though you had one guy ready to go, you could have created a narrative around why the other mook got to go first - and that would have been a more consistent and fair way to adjudicate your initiative rolls.
The decision to assign the highest roll to the guy who was about to shoot may have further incensed the players and made them feel they didn't get the opportunity they should have. It doesn't mean what you did was wrong, but it may have contributed the feelings at the table. It may not have, but I think it's something you should be aware of and consider when rolling your initiative.
Best Answer
Short answer: Yes, you can take a reaction before your first turn in combat.
Reactions and bonus actions aren't ever something you “have”, they're something you do. And there is a limit on the number of times in a round that you can do them.
As a consequence, it doesn't make sense to ask if you “have” either one when you are surprised. What matters is whether you can do things that count as a reaction or as a bonus action. The only reason you wouldn't be able to take a reaction before your first turn is if you're surprised (both quotes from PHB page 189):
So since you're not surprised in this scenario, you can still take reactions.
Also note that surprise also doesn't occur just because you lose initiative. To be surprised:
If you're used to earlier editions of D&D that have a “flat-footed” rule that make you vulnerable in a surprised-like way until your first-round initiative comes up, that might be the source of your assumption that someone can't act in 5e until the first time they have a turn in a combat. There is no similar rule in D&D 5e: if you're unsurprised, you are no more vulnerable before your first turn than after.