The lamp power factor can be calculated by the same formula you used in reverse, ie when you know I and find out PF. It consists of a little displacement and a lot of distortion, so cannot be calculated just from phase angles like a purely displacement PF
It can be numerically measured if you have both Ilamp and Vlamp on a scope or power analyzer with some sampling and processing
HID ballasts are fairly efficient, so R is not significant compared to X. However, Vdrop(R) is almost directly in phase with Varc. To include it, change "Varc" to "Varc + Ilamp * Rbal" in the right places
Something im thinking about now..
In my recent visit to Dor -
https://www.lighting-gallery.net/gallery/displayimage.php?album=9118&pos=0&pid=264621We lit (for the first time since i found it) the M4A3 as is with the lamp and gear that came in it
Lamp is IIRC Philips Belgium SON-T Master 250W. Ballast is ELT Spain standard 250W ballast. Ignitor is ELT ballast dependent ignitor (connected to the ballast tap, not adding resistance in series with the lamp in run mode)
The nominal current for it is 3.0A
In our run up (measured with a fairly basic Uni-T AC+DC clamp meter, hall effect based), the current went from 3.15A for starting (seems legit, tho i expected more) to under 2.6A (quite a bit below the 3A)
Line voltage was around 220V, gear for 230V