I double checked everything, went through every possibility for a column. I've got the Company column selected, and it's obviously correct.
There's a disconnect on the two machines in our firm that are using this software so far, both were installs on-premise. My setup is as an Administrator and the other computer is not. Both have the same blank Company column in My Open Leads or All Leads view.
Leads field mapping shows that the field in data entry is called Company Name. So it seems mapped correctly. Maybe erase that field and put it back, to try to reset it?