Impersonal "se" and passive "se" work almost the same way, except for what you have already mentioned. So yes, your summary is perfectly correct. The difference between both is that the direct object of the active voice sentence becomes the subject in passive "se" sentences, while impersonal "se" sentences don't have a subject because, as you correctly assume, the active version contains an intransitive verb (and in Spanish, unlike English, only direct objects can be the subject of a passive sentence).
Consider:
Aquí la gente habla español. Subject = la gente; verb = habla; direct object = español.
Aquí se habla español. Subject = español; verb = (se) habla, where "se" is a reflex passive marker or "marca de pasiva refleja".
You can tell "español" is the subject because the verb "habla" must agree with it. If you change it with a plural Noun Phrase, you have to pluralize the verb too:
Aquí se hablan varios idiomas. Subject = varios idiomas; verb = (se) hablan, where "se" is a reflex passive marker or "marca de pasiva refleja".
In impersonal "se" sentences, the verb can never be pluralized. Compare:
En el avión la gente no fuma. Subject = la gente; verb = fuma; there is no direct object
En el avión no se fuma. Subject = there's no subject; verb = (se) fuma, where "se" is an impersonal sentence marker or "marca de impersonal"
Since there's no direct object in the active sentence to start with, there's no subject in the passive one. You can easily tell one from the other by looking at the verb. If it is transitive and there's something that looks like an object, that's probably the subject, and you have to conjugate the verb accordingly (passive sentence). If the verb is intransitive, then it must always be in singular (impersonal sentence).
An exception to this rule is when the direct object is a person and is introduced by "a". From the RAE:
Si el elemento nominal expresa persona y no va precedido de la preposición a, se emplea también la construcción de pasiva refleja:
Se buscan actores para la película.
Se contratarán nuevos trabajadores para el proyecto.
Se necesitan especialistas en informática.
Si el elemento nominal expresa persona y va precedido de la preposición a, debe emplearse la construcción impersonal; por tanto, el verbo irá en singular aunque el elemento nominal sea plural:
Entre los gitanos se respeta mucho a los ancianos.
Se entrevistó a los candidatos para el puesto.
Se busca a quienes presenciaron lo ocurrido.
In SpanishDict, all the sentences below are said to be impersonal:
- Se habla español en Uruguay. (They speak Spanish in Uruguay.)
- ¿Cómo se va al teatro? (How does one get to the theater?)
- Se escribe “pastel” p-a-s-t-e-l. (You spell “pastel” p-a-s-t-e-l.)
- ¿Cómo se dice “table” en español? (How do you say “table” in Spanish?)
- Se dice que va a llover (They say it is going to rain.)
You can tell some of them aren't actually impersonal by changing the Noun Phrase. Consider
- Se hablan varios idiomas en Uruguay. (They speak several languages in Uruguay.)
- ¿Cómo se va al teatro? (How does one get to the theater?)
- Se escribe “pastel” p-a-s-t-e-l. (You spell “pastel” p-a-s-t-e-l.)
- ¿Cómo se dicen estas palabras en español? (How do you say these words in Spanish?)
- Se dicen muchas cosas (They say a lot of things.)
As you can see, the verbs in the first, fourth and fifth sentences changed so they would agree with the subject (the italicized parts), so they aren't actually impersonal but reflex passive sentences. The second one is impersonal because "ir" is intransitive, and the third one is tricky. I would call “pastel” p-a-s-t-e-l a complemento circunstancial de modo, or maybe a complemento predicativo, but not a subject. Had the same sentence been worded as “Pastel” se escribe p-a-s-t-e-l, then it would have been more evident that Pastel is the subject, and the sentence could have been classified as a reflex passive sentence as well.
What is curious about that page is that practically the same sentence ("Se habla español aquí" and "Se habla español en Uruguay") is classified first as an impersonal sentence and then as a passive one. Maybe it was just a slip. The rest of the information from the same page is correct.
For a more detailed explanation of the uses of "se", check out this entry on the RAE (only in Spanish).
Edit: I just checked out the other two sources you mention. Wyzant got it right except for when they say "The pronoun “lo”, “la”, “los” or “las” may replace the direct object. Example: “Aquí se lo habla” (el Español)". That's utterly wrong. Under no circumstance can a passive sentence take a direct object; what they call direct object is actually the subject, and of course "aquí se lo habla" is nonsense. SpanishGrammarGenius is both concise and correct.
A key concept here is that just because a verb can take a direct object it doesn't mean it's working transitively, so to speak. A good example is given in the first page:
- “En Colombia se baila mucho la Cumbia.” Since “la Cumbia” is a direct object of the verb “bailar”, this sentence is in the passive voice.
- “En Colombia se baila mucho.” Since there is no direct object for the verb “bailar”, the sentence is an impersonal one.
The fact you decide to call the first sentence impersonal or passive is unimportant: if the only difference is verb agreement, which only makes a difference when the subject is plural, and the subject is absent when the verb is working intransitively, whatever you decide to call it won't have any implications, neither grammatically nor pragmatically.
