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.

Yes

Ready action requires (emphasis mine):

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.

The general rule is 1 Action per turn, but then specific rules can override that (e.g. Action Surge, Haste, etc.). I believe that Ready is another specific rule that overrides this as well because while you are using your Action to Ready, what you are really doing is using your Reaction to move your Action to another time.

Attack vs. attack (or does action have a meaning here?)

Upper case and lower case a have a meaning in the PHB, with the former being the Action described under Actions in Combat in the PHB.

When Ready asks you to choose your action, you use the actions listed there. The cases described in the PHB are "Use an Object" and "Movement" for the Ready (since you can only do one.)

In the case of this question, the Action that is being taken is Attack.

Attack Action on your turn

If you take the Attack action on your turn, then it comes with all the fun stuff that happen on your turn. This includes Extra Attack (which typically doesn't happen on a Ready whose trigger occurs NOT on your turn.)

In contrast, you can look at another reaction: Opportunity Attack. In this instance, the language does not say to take an action but to simply make a melee weapon attack. The difference is in the use of Action in the phrasing for Ready.

Actions vs attacks

The language in Ready is to choose the action. Actions are a defined term in the rules under Combat->Actions in Combat.

At other times, there is other language used when specifying melee/ranged/melee weapon/ranged weapon attacks. The language in Ready does not say to Ready an attack (lowercase a), it says to ready an action.

There are also several referenced examples of Ready Action and Extra Attack/Multiattack that reference the On your turn as the important qualifier. Had they not intended a Ready attack to be the Attack Action, then the discussion would have ended there and not referenced the turn requirements.

There are also other specific examples of more than one action on a turn that override that general rule: Cunning Action, Haste, Action Surge, etc.

Intent Support from Jeremy Crawford

The action choice intent (emphasis mine) is also provided by Crawford:

The Ready action lets you ready any action you can take, including Attack, but Extra Attack is on your turn.

Not only has he clearly stated that you take an Action, and that Action can be Attack (uppercase A), but that Extra Attack only functions on your turn.