The only measurement that is not a convention
Inner diameter in millimetres is the real measurement. Every national ring size is a label laid over it, and the labels do not agree. The US scale defines size 0 as an inner diameter of 0.458 inch with each whole size adding 0.032 inch. The European or ISO size is simply the inner circumference in millimetres, which makes it the most sensible of the lot. Japanese sizes follow JIS: size 1 is 13.0 mm across, each size adding a third of a millimetre. British and Australian sizes use letters, anchored at US 5 equalling UK J½.
Because these are conventions rather than derivations, published charts genuinely disagree by up to half a size, and two shops can both be right. If a ring is being cut, cast or resized rather than tried on, give the workshop the millimetres.
Measuring without getting it wrong
Measure at the end of the day and at normal room temperature. Fingers swell in heat and late in the day and shrink in cold and first thing in the morning; the swing across a single day is commonly close to half a size. Measure more than once, on different days if it matters.
The knuckle is usually the real constraint. If it is much wider than the base of the finger, the ring has to be sized to clear the knuckle and will then spin — sizing beads or a fitted bar solve the spinning without fighting the knuckle. Wide bands need more room: anything over about 6 mm across sits on more of the finger and feels tighter at the same measurement, so a quarter to half size up is the usual allowance.
String and paper strips both compress and neither slides over a knuckle, so they tend to under-read. Treat them as a starting point and confirm with a proper sizer before anything is cut. And note that full eternity bands, tension settings and most titanium, tungsten and ceramic rings cannot be resized at all.