Here's an animation of a toned Peace Dollar (not mine) photographed with 'pseudo-axial' and 'straight' lighting techniques:
Even as it is, I vastly prefer the straight shot. I think its slightly underlit, but I think if it were properly lit there would be absolutely no question that's the superior picture. I understand that axial lighting techniques can have a place for certain coins, but in my opinion those circumstances are rare.
Ray, what do you mean by "pseudo-axial"? I know you are very exact in your description of photography methods (which I much appreciate) - but more details about what you mean by the "pseudo-" prefix in this case? In general, I hate the way that axial photos make coins look so flat and lifeless. Sure, it makes all the colors pop (sometimes in a cartoonish way), but I think straight on photos should always be the primary photos for a coin when selling or auctioning - and if one wants to show other views, those should be secondary.
I've never had much use or luck with axial photos. Considering most results I've seen I think it's more of a time waster.
With axial lighting, the coin and camera are in the same orientations as what I am calling "straight" shooting, but a semi-reflective mirror is placed between the coin and camera, and light is reflected down to the coin from the side. The true effect of this is that light is shining directly from the source, off the coin, and up to the camera. This effect can be created by tilting the coin toward a light source such that the light reflects off the coin and into the lens similarly to axial, so I call it "pseudo-axial". Axial or pseudo-axial photos have no cartwheel luster, but they do bring out the colors in toned coins, especially proofs. In general I agree with Bob (robec), and have never been successful at a true axial system in that I did not like the results, but every so often someone asks me to set up a system with axial light to bring out those "deep" colors that only this type of lighting will produce. A good justification for axial/pseudo-axial photos is that this is how most folks hold a coin when viewing in-hand, so it's really the only way to get the "in-hand look" that is so prized.
This is why I ended up doing both for the same coin, to compare the two styles. The animation shows on this coin that there is not a lot of benefit to axial-style lighting, but for sure there are colors that don't show up in the straight shot. The animation also is similar to how one would hold a coin and tilt it toward and away from the light, so I thought it would be good to have both together.
I think the Coin surfaces should be as close as possible to parallel to the camera lens for consistent focus across the surface. When you tilt the coin in relation to the camera lens that can cause focus issues and require a wider focal plane.
Ray, I was in no way judging your photos. I was making a general statement about the danger of axial-style images. In my opinion, you have done more to advance coin photography than likely any other person on the planet - Mark Goodman included. You have helped innumerable people both here and on CCF - I consider you a mentor and I know you have also advanced forward the photography of @robec and others! Great demonstration of differences in coin looks. My example - I used to own the Barber half below (sold in 2018), showing both the straight on and axially lit views. The coin looks more like straight on in hand at most angles. The axial images look like the coin in hand at about a 5 degree window of views. I consider the straight on images a much more honest view of the coin for display and selling.
I am a big fan of direct (straight) lighting but axial works well with toned proofs. Since the coin in question is a toned business strike, I strongly prefer the straight lighting.