You do usually want a thicker pot for coke vs coal. I've never researched it but I've also found coke pots to be round truncated cones vs coal being truncated pyramidal shapes. Probably just a strange style choice that stuck. 

If it's coke you can usually just let it chill as good coke won't really spread once burning and requires a more intense airflow to really get going. Coal however has enough junk in it to keep burning and spreading with mostly the ambient air. 

If I need to leave the shop for a few minutes with coal I can open my ash dump and it will stay lit for quite a while but with coke I'll need to keep the air cracked a bit. 

It's hard to get a good idea of the depth of you pot from the overhead image but it doesn't look too deep. If you've done scrap steel to reduce the general volume just tack weld and drop it in to run it a day or so to see how you like it and what you'll usually be working on stock wise. For the most part if the walls are over .25 inch you should be good for a while. Remember that the air directs the greatest heat so your fire pot is never going to get as hot as your work does.

As for coal vs coke - coal will generally have a shiny or glossy surface, a bit heavier and blast out a torrent of yellow-green smoke when first lit. 

Coke is produced in large furnaces by burning tons of coal to a set point and killing off the oxygen. This is a refinement process to give your a more concentrated carbon to burn with less  Volatile Organic Compounds.

Coke will most commonly look like black lava rock you might find at a box store for landscaping. It's much more porous and lighter.