A trap and a vent serve two completely different purposes.
The trap allows condensated steam (water) to return to the boiler, while preventing the steam from passing through. It must be located at the very lowest point on the radiator. If any water is trapped in the radiator you can have steam hammer. A trap will also let air out of the radiator when the steam is allowed in. This is why no vent is necessary on a two pipe system like yours. However, if the manufacturer recommends one, then you need one.
Vents are only required on one pipe steam systems. The vent allows the air that is inside the radiator to be forced out so that the steam can occupy the space. The vent is a valve that allows air to leave, but closes when the steam has arrived. The vent is usually located at the far side of the radiator from the steam/condensate pipe, and usually but not always, near the top. This is so the steam will pass through all the tubes before arriving at the vent, forcing out most all the air.
. All vents and traps have rules about how they are oriented. I could not tell from your photo what the top was, you said it was a vertical radiator, but rotate the photo 90 degrees. That would make it a horizontal radiator.
You have not stated what your problem is with the radiator. If it is not producing enough heat, it is more likely the trap is improperly selected or has failed, or you have the incorrect steam pressure.