The text under the simulation distance selector in the new UI describes each simulation distance in terms of an "N x N block radius" (e.g. "64 x 64 block radius" for simulation distance 4). This wording is nonsense: a radius a one-dimensional measurement; "N x N" describes an area, not a radius.
Moreover, an N x N area is much smaller than an N-radius area:
64 x 64 = 4906
3.14159 * 64 ^ 2 = 12,868
Finally, assuming the intended meaning is "a N block radius", the full text is not strictly correct. "Loads and applies changes within a N block radius" is not how the game works and does not make explicit the connection to the simulation distance being chosen.
A more accurate description would be "loads and applies changes in chunks that come within a (simulation_distance x 16) block radius." However, if the mention of "chunks" and multiplying by the number of blocks in a chunk seems too technical, it could be phrased obliquely as "loads and applies changes based on a N block radius."
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I responded to this in MCPE-168049 before I realized you'd updated this ticket with a briefer summary of your comments there (and before I remembered that Mojang was tracking this ticket, not that one). I will briefly summarize my comments here as well.
"Range" is not necessarily a linear quantity; it also has an areal meaning, as in "the range of African elephants". The formulation "N × N blocks" employs the latter meaning.
The simulation area is neither rectangular nor circular. Thus, neither "a range of N × N blocks" nor "a horizontal range N blocks from the player" is accurate. But the true shape has no geometrical name and is too complex to describe in a hint, so I would agree that your suggestion is better than the current hint.
In the other ticket, I suggested "within roughly [perhaps 'approximately' would translate better] N blocks of the player" to make it clear the description of the area wasn't meant to be precise.
Crucially, even if we assume Mojang chooses to retain the "N × N blocks" formulation, the hint currently understates the true size of the simulation area by a factor of 4. The numbers would need to be doubled. (GoldenHelmet, I realize this is equivalent to your last bullet point above. You didn't mention it in the other ticket, but I did, so I'm reproducing my additional comments on it here.)
Since this ticket wasn't closed after the hint was updated and the other ticket was never submitted to Mojang, I intend to close MCPE-168049 at this time.
In 1.19.80 Previews the "an N x N radius" was changed to "a range of N x N" and "blocks" was changed to "chunks", but this was no improvement--see MCPE-168049.
In 1.20 "chunks" has been changed back to "blocks", but the statement is still nonsenical and misleading in the ways I originally explained:
"range" is just as much a linear/one-dimensional quantity as "radius", while "N x N" describes a two dimensional area.
The simulation area is much larger than "N × N blocks"
A concise and accurate statement would be