I am familiar with trunk ports (switchport mode trunk) and access ports (switchport mode access), but what is the proper term for a port that does not have a
switchport mode command? It wouldn't be a trunk port or a switchport so do you just call it a port?
You could call it either an interface or port.
You will sometimes hear people try to create distinct meanings for the two terms. For example that "port" is hardware (i.e. where you connect the cable) and "interface" is software. Or perhaps port is L2 and interface is L3/routed.
The reality is that to my knowledge, the terms "port" and "interface" really do not have a clear distinction industry wide. When and how they are used in my experience seems to depend largely on personal preference and background (i.e. which vendors/platforms you are familiar with, environments in which you have worked, etc).
Cisco of today generally seems to try to use the term "port" when referring to the "hardware entity" and "interface" when referring to a "logical entity" such as you will find in this example. However in the past they have used the two terms a bit differently, such as the use of port in CatOS. When it comes to calling something a "trunk port" or an "access port", you are really calling out the difference in the mode of operation.
There is only one context I can think of in which I know I have always seen a very consistent use of these two terms industry wide. When referring to a passive connection point (such as on the wall behind your desk or on a patch panel), always use port and never interface. However, to keep it confusing, passive connection points may also be referred to as a jack, socket, or outlet as well.