I doubt your issue is with the water pump.
Found this
back port = port closest to the engine block.
What this means is your heater core should not see heat (or water flow) until the engine has warmed, and not before.
In the picture of your thermostat, at the base there is a second spring. I believe this spring prevents water flow through the heater core while the engine is cold. I can't tell from your picture if that second smaller spring has contracted or expanded due to heating. I assume it supposed to contract with heating. The main spring has obviously contracted to allow water flow to the radiator.
I would measure the plate at the bottom at hot and cold to see if that plate if moving. It should be. If it isn't, replace the thermostat.