Source impedance DOES affect the amount of energy moving in and sloshing
around in a transmission line. It DOESN'T affect the ratio of forward to
reflected waves, and therefore DOESN'T affect the SWR.

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But it DOES affect the indicated SWR and so the indicated SWR is incorrect.

It is the meter which is at fault ! It is designed to indicate correctly
only when the source is 50 ohms.

Here's the proof - Rho = (50-Zt) / (50+Zt) - which you may have seen
before.

SWR, of course, is calculated from Rho and the meter scale is calibrated
accordingly.

If the source is not what the meter expects then it gives the wrong answers.
And its faithful worshippers believe it!
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Reg, G4FGQ