I promised you a few more measurements on 8kHz
See here the difference
The differences are not very large, but you can see at 8kHz sample the values are a bit more towards what they should be. Still more than 5% off though.

I don’t know what the best algorithm for calculating rms is, and if it is true rms or not, but I assume you chose the best one.
That I don’t have the in-depth knowledge is probably the reason why I don’t understand that you sample at the 2 zero crossings of the signal.
Isn’t it sufficient enough to calculate a running average (of the absolute values of course).
Lets say calculate the average of the last 256/512 samples every 64 samples.
Or average 64 samples and use it as 1 value in the running average.
(an 8 value running average should give the same result as above, but probably with less processor time)

About equipment again:
watch out with using scopes as reference as well. Digital scopes tend to have an DC accuracy of 1.5 – 2.5% + some zero error. (even the high end ones)
I would recommend to search for a 6.5 digit bench DMM.
Most of the times those are accurate and have a low drift over the years.