Yes, that's correct. The phases of the Moon are defined in terms of the Moon's elongation: the difference in (geocentric) ecliptic longitudes of the Moon and Sun, with New Moon at 0°. The ecliptic latitude is ignored.
From Wikipedia Lunar Phases
There are four principal (primary/major) lunar phases: the new moon, first quarter, full moon, and last quarter (also known as third or final quarter), when the Moon's ecliptic longitude is at an angle to the Sun (as viewed from the centre of the Earth) of 0°, 90°, 180°, and 270°, respectively.
— Seidelmann, P. Kenneth, ed. (1992). Explanatory Supplement to the Astronomical Almanac
However, that article then contradicts itself by saying
Each of these phases appears at slightly different times at different locations on Earth.
which doesn't make sense, given the geocentric definition stated by Dr Seidelmann.
Here's a plot, created using JPL Horizons, of the Moon's ecliptic elongation for the first synodic month of 2022 (with a 6 hour time step), taken from my answer to a similar question.
As you mention, to calculate the illuminated fraction of the Moon we need to use the true 3D Sun-Moon-Earth angle. Horizons provides that angle, but in the range 0° to 180°. Here's the corresponding plot, over the same time span as the plot above.
To compare these values we can transform the elongation angle by subtracting it from 180° and taking the absolute value. When we subtract the Sun-Moon-Earth angle from the transformed elongation angle, we get a plot like this (for 2021, with a 1 day time step).
The difference is usually quite small, except near the New and Full Moons, when they can differ by a couple of degrees. I assume the variation from month to month depends on how close the Moon is to the ecliptic at New & Full Moon, but I haven't investigated that.
All of these plots were created for a geocentric observer. The actual observed angles for an observer on Earth's surface will be slightly different. The Moon is relatively close to the Earth, so parallax effects are relatively large. And of course we should also make adjustments for atmospheric refraction, especially when the Moon is near the horizon.