I have made a ton of them from cottonwood, ponderosa pine, stacks of birch circles, and others. I would never do another birch bark as it never lays flat and is too thin to be worthwhile.
The p pine is very red and has a regular pattern of cells lined in a deeper red... Very good stuff; stable, plenty thick, pretty, lightweight, and easy to work with.
The cottonwood is lighter in color, tending toward gray and light tan. It is thicker and very easy to work with and makes a very nice lightweight grip. I do not hesitate to use them.
My lathe is huge and I can gently hold the entire blank section for turning the handle so I skip the entire mandrel glue-up step and it is a huge timesaver... simply ream or bore the pieces to fit the blank and epoxy them in place. Some of my cottonwood bark is literally 2" thick and it gives me a lot more leeway to use whatever thickness rings I want...
There is a corkbark pine that grows in AZ and I would like to get some of its bark to play with sometime...