It seems that I have mis-stated the name as Thunderbt 1. The computer documentation only states "thunderbolt".
The Thunderbolt ports on the 21" Late 2013 iMac are Thunderbolt 1 ports. They are just called "Thunderbolt", but TB1 indicates that they are 10Gbps ports.
Back when the Late 2013 iMac came out, people just called them Thunderbolt ports, but when Thunderbolt 2 came out a few years later on Macs, people commonly state if they are refering to TB1, TB2, or up ports.
The connector on TB1 is the same connector that is on TB2, It is also the same connector on Mini DisplayPort.
IT IS NOT SQUARE!!!! It may look like it is, but it is the same shape as Mini DisplayPort.
The 21.5 Late 2013 IMac has two Thunderbolt ports, both being Small square dimensions. I have tried both a mini display port and a USB-C cable, following direction of documentation supposedly for this computer but apparently for the 27 inch system. The plugs of the cables I tried are too large and are not square
USB-C is a different connector, and will not with with Thunderbolt (1 and 2) without additional adapters and equipment. Definitely overkill for what you are trying to accomplish.
Mini DisplayPort has the same exact connect as TB (1 and 2). Neither Mini DisplayPort or TB (1 and 2) are square.
If you have a Mini DisplayPort to HDMI cable/adapter, it will definitely fit into your iMac (assuming you actually have a Late 2013 iMac). It can only go it one way, and not reversible like USB-C and Lighting connector.
If you have to, turn your iMac all the way around and see the TB port so you know what way to put the Mini DisplayPort connector into it.