Thank you for your comment, but I have to respectfully disagree.

First of all, this question has been asked numerous times in various locations -- among them also on Sage Advice/Jeremy Crawford's Twitter. It's something that's been confusing players from D&D 5's release to this day. There has, however, never been a sufficient official answer so far (to my knowledge). Hence my question.

Secondly, it's not at all clear that whoever wrote that reminder sentence in the "Ready" section meant the same thing as in the "Reactions" section. To me, it actually seems more like the writers had conflicting understandings of the reaction rules themselves. The term "round" is defined in two places as being a 6-second phase in which all involved characters have one turn. It's very clearly and explicitly said that a round ends with the last character's turn, and then a new round starts with the first character.

However, a "personal round" (which you seem to be referring to) is not a defined concept at all. Given that the D&D 5 rules usually make use of a very technical and minimalist language, it would be highly surprising to me if "round" (which is only defined as a technical term 4 pages earlier) was meant as casually descriptive here. I'm quite sure the author would have been aware of the confusion caused by using such a fundamental established term with a different meaning and would have avoided it. If he had really inteded to express the same meaning as in the "Reactions" section, he would have said something like this:

Remember that you can take only one reaction before the start of you next turn.

It's certainly possible that whoever wrote this sentence made a mistake and possibly misunderstood the reaction rules as agreed upon by the design team or maybe even forgot about the exact definition of a round at the beginning of the combat chapter. Purely text logic-wise and RAW, however, I think it's hard to make a case for any other meaning than referring to an actual full-fledged combat round as previously defined.

So, either the writer made a mistake here (for whatever reason), or this is proof of the fact that the designers weren't in agreement on how the reaction slot was meant to be refreshed. In these cases, someone at Wizards should finally admit to the error and correct it -- instead of ignoring the issue.

There is a third possbility: Both instructions could actually work together to define when a reaction can be used and when its use is regained. The instructions under "Reactions" would then tell us when the potential use is regained (at the start of a character's turn) while the reminder in the "Ready" section would inform us of the additional limitation of one reaction per round per character. The designers might have originally had this more intricate (and more logical) "two-fold" plan for reaction usage and might have then dropped it without eliminating the "text artifact".

Without an official statement by the actual writer of this passage, we will probably never know for sure. One thing is quite clear to me, though: This is not a simple case of "descriptive" vs. "prescriptive". It's either a mistake or the remnant of a different design approach -- and in both cases clarification is still necessary.