In your example there is a chain of causation which you can describe in a multitude of ways.
You and the switch are not monolithic singularities, you and the switch are complex systems, and together you can be epistemologically seen as a system also.
Your brain causes impulses which case muscles in your arm to contract which cause your arm to move in relation to your torso and your finger to move in relation... etc... causing the tip of your finger to rotate a switch about its pivot causing one contact to touch another contact allowing a current to flow through a wire to a light causing the filament to get hot and produce light...
You can slice and dice the systems umpteen ways and break down the complex chain of causation in umpteen ways. i.e. describe this in many different ways. What happened, happened it is what it is regardless of how you analyze it... and always going back to what it was can be helpful.
So yes, you (broadly speaking) did cause the light to go on. But the light had to be there, it had to be operational (not broken), the power had to be on and ready to be connected by the light switch, the switch had to work, etc. all of these are conditions for the light actually going on.
The light switch, and its presence, did not cause you to turn it on... although arguably its presence was a condition precedent to your deciding to do so.
In the subquote of Rand "The nature of an action is caused and determined by the nature of the entities that act" it's best that you not focus on the word "caused" but the word "determined", especially of you take it out of context of the rest of the explanation. Entities act, those actions are caused by entities, how they act is determined by the nature of the entities.
Causation is not "caused", it is a part of reality. Actions are caused by entities.
Also think about what you take to be an action. An action as opposed to a property or a attribute implies some change, a difference of some state of something over time. A thing simply being, i.e. having mass or filling space is not an action. As such simple being does not require a cause, a thing simply is, existence is identity.
Why do we have a require a concept of causation? Only because of change, something being different from what it was. Change only presupposes causes. The law of causation simply links the nature and specific types of changes to the nature of the entities which bring them about. The changes are determined by the nature of the things acting.