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 can activate Action Surge, but it would waste it
The rules on surprise state:
A surprised creature can’t move or take an action or a reaction until its first first turn ends (remember that being unable to take an action also means you can’t take a bonus action).
Since you can't take an action, that means any actions (even if you have 2) and a bonus, since you can't take bonus actions when you can't take actions.
Yes, an unsurprised character may use their Reaction before their first turn
There are two relevant rules in the Player's Handbook, emphasis added for relevance:
So between these two sections, we glean that the only point where a Reaction is "refreshed" is at the start of a character's turn; there's no explicit point where the "initial Reaction" is granted, therefore it is available before the character gains its turn.
The only scenario where a character would not gain their Reaction is if they are surprised, or otherwise under an explicit effect that denies use of a character's Reaction.
A Character needs an action that allows them to use their Reaction
Most regular uses of Reactions are unaffected by occurring before the first turn a character takes (Attacks of Opportunities, Counterspell, etc.), but one common use is the Readied Action; per the rules, a character won't be able to react with a Ready Action, because they weren't able to set it up in the first place due to not having had their first turn yet.
Strictly RAW, "Actions in Combat" cannot be taken outside of Initiative order. Many DMs waive this restriction on a case-by-case basis because adjudicating this rule for every possible decision a character could make would be cumbersome, but as a general rule, allowing characters to Ready Action outside of combat, as a general rule, can lead to characters getting extra attacks, and is a concept that's meant to be encapsulated by Initiative order.
So unless your DM rules otherwise on the Ready Action, a character would not be able to use the Reaction granted by Ready Action before their first turn.