You said "if we use the agent, the sentence becomes passive voice". I have to disagree. In every one of your examples, the subject of the verb is the patient of the action. Regardless of whether the agent is mentioned, those statements employ the passive voice.
You're asking how to distinguish something that doesn't need to be distinguished. The question of passive vs. active voice is completely different than the question of stative vs. dynamic verbs -- in the same way that the question of how curly your hair is happens to be completely different than the question of how dark your hair is.
In an active voice construction, the subject causes the action. In a passive voice construction, the subject receives the action. It's easy to see that all of your sentences are passive voice constructions because its easy to find the active voice equivalents:
Someone parked the car.
Something increased your work volume.
Someone arrested him.
and so on.
Although the stative vs. dynamic distinction exists in English semantics, it isn't present in English grammar. One rule of thumb is that stative verbs do not work well when employing the continuous aspect. For example, "I am knowing the answer" is a poorly-formed sentence because the verb to know is stative. It seems that every one of your passive voice examples uses a dynamic verb.
One car was being parked outside the gate when I left.
Your work volume is being increased during the holidays.
He was being arrested while I calmly walked away.
and so on.
A stative verb can be used in either the active or the passive voice:
My manager knows me to be an excellent employee.
I am known to be an excellent employee.
A dynamic verb can be used in either the active or the passive voice:
I parked the car outside the gate.
The car was parked outside the gate.
So, how do you distinguish between passive and stative? You don't. You distinguish between active and passive. You distinguish between dynamic and stative. You do those two things separately. They are not related to each other.
Now, as to your follow-up question, let's use the dynamic verb to park in the passive voice and indicative mode:
The car was parked -- past tense, indefinite aspect
The car is parked -- present tense, indefinite aspect
The car will be parked -- future tense, indefinite aspect
The car had been parked -- past tense, perfect aspect
The car has been parked -- present tense, perfect aspect
The car will have been parked -- future tense, perfect aspect
Does the meaning change? Yes. Aspect carries a different kind of meaning than tense, but both aspect and tense carry meaning. Tense has to do with location in time. Aspect has to do with a relationship with time. Often tense and aspect are taught as a single, combined thing, but I find them easier to understand when they are examined as separate properties.
Changing the aspect does not change the tense. Changing the aspect does not change the voice. Changing the aspect does not change the nature of the verb, whether stative or dynamic.
Because the perfect aspect has an explicit relationship with time, it makes the most sense when the time frame is also made explicit, as in "The car has been parked there for several weeks" or "The car had been parked there for a month before anyone noticed it."
Whether a verb is stative or dynamic has very little impact on English grammar. Certain verb forms don't make sense for stative verbs, and that's about it. The properties of voice, tense, aspect and mode have a great deal of impact. Changing any one of those requires changing something about the verb's form.
Let me see if I can clearly summarize all of that.
- Which of the example sentences are passive voice? All of them.
- Which of them are indicative sentences? All of them.
- Which can be interpreted as having a participial subject complement? All of them.
- Which are stative? None of them.
Most of those are unrelated items. One pair is an identity. The passive voice construction is a subset of the participial subject complement construction.
Best Answer
Your sentence is grammatical. "Being well-treated" is an introductory participle phrase used as an adjective to describe the subject of the main clause (she).
That being said, it is a little bit difficult to figure out for certain what you actually mean. It's not entirely clear what "being well-treated" would have to do with getting home sooner. If someone said this to me, I would guess that "well-treated" is supposed to mean that she received fast or expedited service, but that's not necessarily what it means, and you might not mean that at all, so it's ambiguous.
I would suggest being more explicit with something like:
If you want to keep the idea of "well-treated" in there, it is possible with something like:
"Expedited service" sounds just a little stiff, but I can't guess at ways to improve it without more context. Although more context might also make it clear that no causal relationship is implied here at all.
Edit:
When speaking about medical treatment, you would not usually talk about "expedited service," of course. To clarify, there's nothing wrong with introducing a sentence with "being well-treated" or just "well-treated," whether you're talking about customer service or medical treatment.
The problem that causes ambiguity here is the meaning of your main clause, because good treatment doesn't usually imply fast treatment or explain in any way why she got home sooner. However, using a different main clause as an example, both of the following are logical and grammatical!
Note that this also makes sense:
Now it is implying that the very excellence of the treatment caused her to recover faster, rather than implying there was something fast about the treatment itself.