> now how does standard meters calculate AC?
ANY way they want to.
The most basic passive multi-meter (now obsolete) is a mechanical DC current meter. To read AC it uses a rectifier and a resistor. This actually reads "Average". But the scale is calibrated 1.111 times higher, so that the RMS of a sine-wave reads correctly. This actually gives a pretty-good approximation of "RMS" for most "reasonable" waveforms.
My VTVM uses a peak-to-peak rectifier, then scales it 1/2.828 to read "RMS" of sine-waves. On non-sine waves it gives numbers very different than the RMS.
A thermal-meter reads true RMS over some range. The RMS chips read true RMS if the waveform is not too spiky.
If you know the wave is a sine, any of these is fine.
If the wave is not a sine, you gotta ask: what do you really want to know?
If you want to run heaters, toasters, or incandescent lamps from a non-sine source, you want the RMS. This is the "heating power".
If you want to know if a musical signal will "fit" through an amplifier (without clipping), the RMS is not really important, you want the Peak-to-peak.
All this confusion is one reason why we usually use sine-waves for testing.
Your 'scope on 6V6 grid with BIG signal is correct. The 5-watt amplifier is attempting to put out 10W or 20W, and can't.
If you wanted a clean sound, you would play softer (try 20mV to input) and turn-down (try "5" on the volume control). Now the output should be very much like the input, only bigger, and (at the OT secondary) able to pull a heavy load (8 ohms).