OK, here's the deal. The two pins in question seal off the vent and fuel outlet when the hose is removed. Attaching the hose pushing the pins in, and opening the vent and fuel outlet. The vent, when open, is a one-way vent on all but the earliest tanks. That means it allows air to enteer but does not let fuel and air to slop out. It appears you have the one-way vent. The fuel outlet pin is a simple open-or-closed valve. Push it in and fuel can flow, remove the hose and the fuel cannot flow out.
The most common problem is leakage around the fuel pin (and it is a common problem). If it leaks around the pin, your fuel pump sucks air from the leak path, instead of fuel from the tank.
If you look at the fuel pin, you will see a tiny brass washer surrounding the pin. The washer is retaining a tiny o-ring inside. So, if it is leaking, you need to replace the tiny o-ring. That is done by removing the core plug from the back side and sliding the pin and spring out. Now stick a punch into the washer and rock it back and forth to break the washer free. Now you can pick the o-ring out and replace it. You can re-use the washer. Use a hammer and screwdriver to stake it back in.
There are more o-rings acting as valve seats. Might as well replace both of those while you are at it. Of course you will need all the o-rings and core plugs.
Be aware there are two sizes of the tiny o-ring. The very early tanks had the smaller size.