You'll see both terms used as if they're identical, and in everyday use the distinction rarely matters. But they're not quite the same, and knowing the difference clears up why different sources quote slightly different numbers.
BMR (basal metabolic rate) is energy expenditure measured under tightly controlled conditions: fully rested, fasted, lying down, in a thermoneutral room, often after an overnight stay. It's the true biological floor.
RMR (resting metabolic rate) is measured under less strict resting conditions — you don't need the overnight fast and lab protocol. Because those conditions are looser, RMR comes out slightly higher than BMR, typically by a few percent.
The measurement conditions for true BMR are hard to achieve outside a lab, so most "BMR" figures you encounter — including from predictive equations — are really estimating something close to RMR. The popular formulas were validated against resting measurements. In practice the two are close enough that the labels are used interchangeably.
The BMR calculator uses the Mifflin-St Jeor equation, which was derived from resting energy measurements. Treat its output as your resting/basal estimate — the baseline you build TDEE on by multiplying by an activity factor. The small BMR-vs-RMR gap is well within the normal margin of any prediction.
Don't lose sleep over which acronym a source uses. Both describe the energy you burn at rest, both are roughly 60–70% of total daily expenditure, and both serve the same purpose: the foundation you add activity to. For how that foundation is calculated, see the methodology page.
Nearly. BMR is measured under stricter fasted, fully-rested lab conditions; RMR under looser resting conditions and runs a few percent higher. Most everyday 'BMR' figures are really RMR estimates.
Predictive equations like Mifflin-St Jeor were validated against resting measurements, so the output is best treated as a resting/basal estimate — the difference is within normal prediction margins.