What is the way to understand "ergative verbs"?

I am confused with "paired ergative verbs" and "unpaired ergative verbs".

1. He knocked the vase off the table and it broke.

   Is [broke] in this sentence a paired ergative verb, because we can make the transitive counterpart as "He broke the vase"?

2. He died with his boots on, like any good cowboy.

   Then, is [died] in this sentence an unpaired ergative verb, because we can't make any other counterpart?

3. The black horse ran faster than the other ones.

   Now, what kind of verb is this [ran]? Is this sentence below correct? "He ran the black horse." If correct, is the verb [ran] also a paired ergative verb?

4. As I knew, the verb [die] and [run] are intransitive verbs. What is different between an "ergative verb" and "intransitive verb"? I mean, is there a method to divide the verbs?

I [ergative] broke them [absolutive].

Them [absolutive] broke.

This would be how ergativity works in those languages that have an ergative case: if you remove the subject/agent ("I") from what would be a transitive verb, so you only have the patient/theme/etc. ("them") left, resulting in what we would consider an intransitive verb, then the single argument will be in the same case as the secondary argument in a transitive sentence ("them"), as above. The case by which the agent of a transitive verb is marked is called the ergative case, and the case of the object is called the absolutive case, which is the same case as the subject of an intransitive verb. I believe this normally applies to any intransitive verb, so not only to intransitive verbs with a transitive counterpart:

Them [absolutive] arrived.

Them [absolutive] cried.

This system does not exist in languages like English, which use the accusative and the nominative case, where the subject of an intransitive verb will be in the nominative, just like the subject of a transitive verb. We don't have an ergative case or ergative verbs. So you could call break paired and die unpaired, but only in an ergative-absolutive language: it doesn't make much sense to use those terms for English.

What you describe is mainly intransitive verbs in English where the subject fulfils the semantic role of theme or patient, or at least anything but agent; and some of those verbs have a transitive counterpart, where the subject will be the agent and the object theme or patient, like run and break, while others haven't any, like die.