The question is whether you're talking about literal triggers (i.e. the lever on a gun or gun-shaped tool) or figurative ones. I'll skip you some time on the longer figurative answer and address software development specifically.
Literal triggers
People who pull the triggers of guns are known as a shooter:
Shooter
a person who uses a gun on a particular occasion
"The accused White House shooter was arrested on Wednesday by Pennsylvania state troopers."
When we're talking about gun-shaped items that are not actual firearms, you describe either the person or the action they're performing just as they are contextually known, e.g.
- The carpenter nailed the plank to the floor (using a nail gun)
- The set builder glued the leaves to the tree (using a hot glue gun)
- The baker frosted the cake (using a decoration gun)
There is no uniform way to say "they physically operated the trigger on this device that has a trigger", and there's usually no need for such a generic term anyway, given the wide range of contexts in which a physical trigger could be used.
Software development
For software development specifically, "triggerer" is a really bad name, because it is as vague as can be. Software is a massive sequence of tiny events, and each event is the trigger for the next event. You could literally call anything a "triggerer" (or whatever word you'd use).
Classes should be named after their purpose. If the only way to describe their purpose is to say "this class triggers the next thing", in software development terms that's the same as saying "this class doesn't do nothing".
If you consider that this new entity does not invariably set this chain of events into motion (e.g. it's based on a conditional), then you shouldn't name it for its trigger-like (or trigger-wielding-like) nature.
This is too vague to give you a specific answer to. Maybe it's a web controller. Maybe it's a hosted service. Maybe it's a validation rule. Name it after what it is.
But in all cases, you're going to have to focus on a more meaningful role description other than "it doesn't do nothing".
Figurative triggers
This is one of those principles that is encapsulated by an old joke.
Lady: I do say, this train ride has been nothing but bumpy and horrible.
Conductor: It's always bumpier in the last carriage, ma'am.
Lady: Well they should remove the last carriage then!
The joke works because "the last carriage" is a shifting goal post. Even if you remove it, the carriage that used to be second to last has now become the last carriage. Sorry for ruining the joke by explaining it, but there is a purpose to me doing so.
The label of "trigger" is a similarly shifting goalpost. When you understand that B was the trigger for A, and C was the trigger for B, then C becomes "the trigger". In fact, when we label something to be "the trigger", we essentially say that this is the highest level on which the sequence of events (leading to A) is guaranteed.
It's like digging a hole. You can choose how deep you dig the hole, but you are free to stop at any time and say "this is the bottom of the hole", no matter how deep you dug. If you start digging again, and dig deeper, you can then again say that "this is the bottom of the hole". There is no concept of a "bottom of the bottom".
Similarly, you say "this is the trigger" when you stop looking for a further causal chain. If you look deeper down the causal chain, then the label of "trigger" shifts with it.
Let's use an example. Note that I'm listing events in reverse order. The effect is listed before the cause, and its cause, and...
Simple explanation of how guns work:
- A bullet shoots out of a gun
- A gunman pulled the little lever on the bottom of their gun while it was loaded
This is why we call that little lever the "trigger", because it is the thing that sets in motion the chain of events that will fire a bullet from this gun. I could've been much more detailed about this chain of events:
- A bullet shoots out of a gun
- The gunpowder in the round explodes and creates a pushing force on the bullet
- The cocked hammer strikes the back of the round
- A gunman pulls the little lever on the bottom of their gun while it was loaded
I could've been even more detailed about the physics and chemistry of it all, but what I really want to point out that we don't call it the trigger because it directly precedes its effect, but because it initiates the effect. Pulling the trigger is what starts the entire sequence of events.
The takeaway here is that whatever is at the bottom of the list is "the trigger", if you consider each cause and effect to be inevitable consequences of one another.
Otherwise, the "trigger" is event lowest on the list that is continuously connected to the final effect using nothing but (reasonably) inevitable steps.
We can also look at more concrete examples:
- Archduke Ferdinand is shot dead
- Gavrilo Princip pulled the trigger of his gun while aiming it at the Archduke.
So there you have it. That literal gun trigger is the trigger for Archduke Ferdinand's death.
But we're in history class, and we're talking about World War I. What the teacher is trying to get across is that there's more to this chain:
- World War I starts
- Archduke Ferdinand dies
- Archduke Ferdinand is shot
- Gavrilo Princip pulled the trigger of his gun while aiming it at the Archduke.
That specific trigger on that specific gun, on that specific day, was the trigger for World War I.
But when we're discussing such things, we're generally not interested in the workings of a gun, but rather the decision made by the shooter themselves.
- World War I starts
- Archduke Ferdinand dies
- Archduke Ferdinand is shot
- Gavrilo Princip intended to kill Archduke Ferdinand to damage the Austro-Hungarian empire
So since this is history class, we're interested in the people, and therefore Gavrilo Princip was the trigger for World War I. How he killed the Archduke is pretty much irrelevant, so the literal trigger is not really being discussed as an essential part in that chain of events. He could've used a knife and it would've still sparked WWI.
But if we're in psychology or social studies, we may focus on another aspect:
- World War I starts
- Archduke Ferdinand dies
- Archduke Ferdinand is shot
- Gavrilo Princip intended to kill Archduke Ferdinand to damage the Austro-Hungarian empire
- The Austro-Hungarian royal house (which Ferdinand was part of) had historically abused its Bosnian population (which Gavrilo was part of).
Therefore, a social studies expert who is discussing the psychology against seeking retribution against your abuser might argue that it was systemic ethnic disenfranchisement by the royal house against Bosnians that was the inevitable trigger for retaliation against the royal house by the Bosnians, which in turn inevitably started WWI.
Just for completeness' sake, not every cause is considered a trigger:
- World War I starts
- Archduke Ferdinand dies
- Archduke Ferdinand is shot
- Gavrilo Princip intended to kill Archduke Ferdinand to damage the Austro-Hungarian empire
- The Austro-Hungarian royal house (which Ferdinand was part of) had historically abused its Bosnian population (which Gavrilo was part of).
- Marija and Petar Princip had a second child.
Even though that second child turns out to be Gavrilo, no one is claiming that Marija and Petar set about an inevitable chain of events that caused WWI.
Well, if you did argue that point, then you would call event 6 "the trigger". But if you aren't arguing that point, but are focusing on social dynamics, then you'd still only call event 5 "the trigger".
To summarize
In every single list I made above (except the last one, for reasons explained there), you would be correct in stating that the last entry on that list was "the" trigger that cause the first event on that list.
Context matters. What you call "the trigger" is based on what you consider to be the start of a chain of events.
Which finally brings me round to your actual question:
it's desirable to distinguish the thing activating the trigger from the trigger itself
You are essentially digging a level deeper, and therefore what you're trying to refer to as the "triggerer" is in fact the (goalpost-shifted) new trigger.