"The purpose is to provide a better photographic rendering of coins. "
Axial lighting is one of the many ways of lighting in coin photography. For some coins (coins with colorful toning, for example) and for certain purposes (show details of the coin), it probably is a "better" way. For those who wish to show a bright field on the coin, this is an easy way to achieve that purpose. But it is not always better, especially when it is used alone, all by itself. Axial lighting is an even, broad light source. As such, the photos would come out "flat", missing shadows which help create a 3-D effect. It cannot bring out the luster band(s) which is vital for business strikes like the Morgan dollar. Circulated coins would appear "lifeless" and bland under this light setup, as your Seated Liberty shows.
There is not general "better" lighting that fits all the coins and all the purposes. Lighting setup needs to match the coin and the purpose. More often than not, different methods of lighting are used in combination, such as axial lighting with direct lighting, as in the second photo in previous my post.
"Adjusting the lighting source is one variable to toggle, I suppose among many. I don't think the light source's field of illumination is an issue given how large the light source is in my compact setup. "
Here is a post that shows several setups of the axial light source: http://goccf.com/t/197120&SearchTerms=axial. For the one that uses a single Ikea LED light, a white screen is place in front to expand the size of the light source. I personally use an LED panel, which can adjust both the intensity and color temperature. Before, I used three diffused Ikea LED lights which took a lot of space. The unevenness of lighting on your Morgan may have two causes. One is that the axial lighting source is too narrow. The other is that the angle of the glass is not quite 45 degrees. I actually often take advantage of the different angles of the glass to achieve some special effect. In the photo below, I left a small dark area on the coin by adjusting the angle, just to create some contrast. All white sometimes can be boring.

I use two 5X7 picture frames connected by a friction hinge. It allows me to adjust the angle freely:

Axial lighting is one of the many ways of lighting in coin photography. For some coins (coins with colorful toning, for example) and for certain purposes (show details of the coin), it probably is a "better" way. For those who wish to show a bright field on the coin, this is an easy way to achieve that purpose. But it is not always better, especially when it is used alone, all by itself. Axial lighting is an even, broad light source. As such, the photos would come out "flat", missing shadows which help create a 3-D effect. It cannot bring out the luster band(s) which is vital for business strikes like the Morgan dollar. Circulated coins would appear "lifeless" and bland under this light setup, as your Seated Liberty shows.
There is not general "better" lighting that fits all the coins and all the purposes. Lighting setup needs to match the coin and the purpose. More often than not, different methods of lighting are used in combination, such as axial lighting with direct lighting, as in the second photo in previous my post.
"Adjusting the lighting source is one variable to toggle, I suppose among many. I don't think the light source's field of illumination is an issue given how large the light source is in my compact setup. "
Here is a post that shows several setups of the axial light source: http://goccf.com/t/197120&SearchTerms=axial. For the one that uses a single Ikea LED light, a white screen is place in front to expand the size of the light source. I personally use an LED panel, which can adjust both the intensity and color temperature. Before, I used three diffused Ikea LED lights which took a lot of space. The unevenness of lighting on your Morgan may have two causes. One is that the axial lighting source is too narrow. The other is that the angle of the glass is not quite 45 degrees. I actually often take advantage of the different angles of the glass to achieve some special effect. In the photo below, I left a small dark area on the coin by adjusting the angle, just to create some contrast. All white sometimes can be boring.

I use two 5X7 picture frames connected by a friction hinge. It allows me to adjust the angle freely:
