4 9/16 looks correct. Sometimes I also write down different Numbers then what I see.

x2.

The choices are 4 9/16", 5 1/4", or 6 9/16".

Those are some of the choices. Many companies will make it to any dimension you give them. I rarely use a “standard“ sized jamb depth.

4 9/16" looks correct to me as well. 6 9/16" is typically for a 2x6 wall so they just made an error and as Todd says it happens.

We do have a vaulted ceiling immediately inside the doorway. Is there an easy way to confirm 6 9/16" or would we have any issues fitting a 4 9/16" into the space if it was in fact framed larger?

You would just need a jamb extension kit or make your own.

I would call and confirm with whoever measured the door and perhaps send the pics you posted. The existing surely looks to be 4 9/16ths. You want to get this right and not mess around with jamb extensions. ideally you want pre finished or cladded jambs from the factory and all 1 piece . It is hard to make a classed jamb look good with an extension and is more work.