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Hi Ed,
Just returned home...don't have an nVNA handy...
There are 2 questions on the table.

1. What does the H4 show for FORMAT > LINEAR?
ans: LINEAR - Magnitude (absolute value) of Sxy.
That's if I recall correctly.
That's why the Linear col in the spreadsheet calculates | S11 |.
The user manual is an attempt to lay out for folks what the H4 is doing...for example, when I first saw LINEAR, I wondered what that could be. I don't associate the word "linear" with "absolute value", "Euclidean norm", or "p-norm", where p=2. So I took a data set, and, using the same data (not scanning), changed the formats to see what they displayed.
I /think/ that if someone fires up an H4 and sets the display format to LINEAR, they will find it is | S11 |.
If that turns out to be the case, then right or wrong, that's what LINEAR means in an H4.

2. What does "one_port.pdf" say?
Well, to begin with, not the point. The point is to document what the H4 is doing, whether or not it is doing it correctly.
Then..."one_port.pdf" says
Just as for the Return Loss, the VSWR (dimensionless) is calculated from rho...(11)
where 1+rho = 1+| S11 |
seemingly suggesting that rho is the magnitude of S11.
I havna really read the doc...

I welcome discussion...