Sat down to practice tonight and decided to muck about with hand and lyre positioning.
I have an article by Graeme Lawson, the English archeo-musicologist, that tries to derive the proper positioning from different representations of "David with his harp" in manuscripts and on carved stones. It's a touch complicated:
Hold lyre vertically in lap or on left thigh, strings facing audience.
Tip ~40 deg to the strap-hand side (the left hand in the iconography, YMMV)
Tip ~20 deg forward, away from you
Rotate through the lyre's long axis so your strap-hand fingers are comfortable on the strings
I tried this a few years ago, when my strap was tied low on my lyre yoke, and I seem to remember it being fairly comfortable. With the leather-supported strap up high, though, this was not a good position. In particular, my fourth finger was having an even harder time than normal articulating cleanly because of the angles of my arm and hand.
With the new, higher straps, resting on my strap-hand thigh, Bagby-style, was perfect. I played - not very long, maybe ten minutes or so - with no fatigue, which was more than I could say for even a few moments of my earlier contortions.
I should probably take another cue from Bagby as far as position relative to the audience goes. Seated like that, you can't have the lyre soundboard or your mouth dead-on to the audience without the other one being 90 degrees off, unless you twist your neck around - not the best way to sing. Bagby looks like he sits at a bit of an angle, with neither his face nor his instrument looking directly out at the audience. And of course, he's very animated when he performs and does sometimes turn his head to catch different corners of the audience.