The temperature of an object is the result of the balance between heat being added and heat being lost. In the case of greenhouses, both effects you described are in play, but stopping the air exchange has a far bigger role than blocking infrared light. More details below:

Suppose you're standing in an open-air garden on a cold but sunny day. The majority of the heat reaching you comes in the form of solar radiation, i.e. sunlight. But because the environment around you is fairly cold, that heat is quickly lost to the cold air.

In a greenhouse, you still have roughly the same amount of heat incoming. But because the air is (mostly) trapped, heat loss is greatly reduced. More specifically, instead of the heat being lost by convection in the previous case, heat entering the greenhouse has to be conducted through the glass walls in order to escape. This is much harder, so the equilibrium temperature inside the greenhouse is higher than the garden.

Side note: regular glass is actually pretty high transmissivity in the infrared. Although the stuff inside your greenhouse aren't at a temperature where they're producing a lot of infrared radiation anyways (I hope).

In the case of Earth, incoming heat is radiation in the visible spectrum, while outgoing heat is radiation in the infrared. Having more greenhouse gases in the atmosphere block more of the infrared heat from leaving, so we end up with higher average global temperature.

So in the broad sense of reducing heat loss, the term greenhouse effect is correct and easy to understand. It's just reducing radiative heat loss (for Earth) vs convective heat loss (for greenhouses).