Several spells—or up-leveled versions of spells—have language such as this (charm person At Higher Levels from the Player's Handbook):

The creatures must be within 30 feet of each other when you target them.

I believe the above statement is equivalent to "The creatures must be within a 30-foot diameter (i.e. a 15-foot radius)."

When playing with such spells and using figurines and mats, I and others have sometimes started to measure distances between pairs of targets, rather than simply dropping a circle template of 30-foot diameter to see whether these can all fit inside.

Am I understanding correctly, and are the two phrases in my title exactly equivalent?

Note 1: I understand the difference between targeting an area of effect, like a circle or sphere, and targeting specific creatures within an area.

Note 2: Similar language appears in other editions, for example mass cure light wounds from Player's Handbook v3.5: "One creature/level, no two of which can be more than 30 ft. apart."

No, "all targets within 30 feet of each other" is a more flexible criterion

We can show that the two criteria are not equivalent with a simple counterexample: suppose 3 targets are each 30 feet away from each other, such that they form an equilateral triangle with side length 30. The smallest circle that can enclose these 3 targets (known as the "circumcircle" of the triangle) has a diameter of of about 35 feet (see here; enter 30 for the edge length and press "Calculate").

However, going the other way, getting all targets inside a 35-foot circle is also not sufficient to establish that all targets are within 30 feet of each other. If two targets are at the edges exactly opposite each other, they will be 35 feet apart.

So, "all targets within X of each other" is not equivalent to "all targets within a circle of diameter X", and furthermore, there is no other diameter that you can substitute for X to make them equivalent.

Additional notes

In my answer, I have treated each target as a mathematical point with zero volume. It is fairly straightforward to extend my answer the targets with non-zero volumes, but it makes the math a bit more complicated, so I've opted to keep the idealized representation or targets as points for the sake of simplicity.

However, I should also note that my answer assumes that you are using Euclidean geometry. The Player's Handbook includes variant rules for playing on a grid, using quite non-Euclidean rules for calculating diagonal distances. As Mindwin's answer notes, these distance rules allow you to simplify the criterion of "all targets within 30 feet of each other". However, if you play on a grid with the variant diagonal rules from the Dungeon Master's Guide (putting diagonal distance at roughly 1.5x cardinal movement), or if you play using hexes instead of a square grid, this simplification no longer holds true.

No. But you can swap the definition with a square if you are using the grid movement rules.

As Ryan proved in his answer, given three targets they have to be in a triangle with 30 feet of edge. But in the game this triangle cannot be drawn. The diagonals count as five foot of distance the same as the straight lines, so the best representation one can make is actually a 7 x 7 spaces square.

The distance to one's own square is 0 feet. The distance to the adjacent square is 5 feet.

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It is easy to prove that all squares inside this grid are within 30 feet of each other. Any square outside this will be more than 30 feet away from the opposite corner / edge.

If each diagonal costs 1.5x the movement (alternate DMG rules) then it becomes more like a circle:

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The cells marked with XX are actually 35 feet of each other. Four of them in opposed sides must be ignored. Thanks Ryan

In the English language, those two phrase are not the same. Mathematically too.

Take for instance the Reuleaux triangle. It would allow all targets to be within 30 ft of each other, but it does not have a uniform 30ft radius. So in terms of pure topography, there can be differences.

And for the language, "creatures must be within 30 feet of each other when you target them", I would also take the more general meaning that targets do not all have to be in the same 30ft space, but only that each target is within 30ft of another target. So you could have multiple targets in a row each spaced 30ft apart. Or four targets on the corners of a 30ft square. Mathematically, the target directly diagonal is more than 30ft, but each adjacent target is only 30ft.

There is also a piece missing, where it could be that the import factor is "when you target them". Perhaps those spells you are looking at are not instantaneous and therefore the targets can move after targeting and they only need to be in proximity at the time the spell is cast. That would also make the meanings different.