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I found lots of examples of both in dictionaries but cannot see any regularity.

Usually choice of an article depends on whether I mean a specific object or a class of objects.

But what exactly is meant when it comes to "right"? Is it ability to do something or a specific case when I do it?

For example, in the sentence:

I have _____ right to vote, because I'm 18.

what should I use?

There is the cite on https://en.wikipedia.org/wiki/Rights from Stanford Encyclopedia of Philosophy:

A right to life, a right to choose; a right to vote, to work, to strike; a right to one phone call, to dissolve parliament, to operate a forklift, to asylum, to equal treatment before the law, to feel proud of what one has done; a right to exist, to sentence an offender to death, to launch a nuclear first strike, to carry a concealed weapon, to a distinct genetic identity; a right to believe one's own eyes, to pronounce the couple husband and wife, to be left alone, to go to hell in one's own way.

All these "rights" are used with "a".

The first result of the search of "right examples" in Google reads:

Some examples of human rights include:

The right to life.

The right to liberty and freedom.

The right to the pursuit of happiness.

The right to live your life free of discrimination.

The right to control what happens to your own body and to make medical decisions for yourself.

How can that be explained?

Either option would be correct and sound just fine.

I have a right to vote because I’m 18.

I have the right to vote because I’m 18.

Sometimes the definite article is assuredly the correct one to use, and sometimes the indefinite article is the correct one to use, but other times the difference is so insignificant that it doesn’t really matter which one you use.

Additional context might tilt the circumstances one way or the other, but, in this case, either one of those is acceptable and they pretty much mean the same thing.

Put another way, if someone is standing in line at the polls, and they say:

I have a right to vote in this election!

or:

I have the right to vote in this election!

no one is going to correct their grammar because they used the wrong article – no matter which version was uttered by the prospective voter.

This would hold true for other rights as well:

We all have a right to pursue happiness.

We all have the right to pursue happiness.

When we say the latter, we are referring to a specific right. When we say the former, we implicitly acknowledge that we are not talking about our only right. But no one will hear the latter and assume it implies we have no other rights.

You can even switch them around in the excerpts in your question with no adverse effects:

The right to life, the right to choose; the right to vote, to work, to strike; the right to one phone call...

Some examples of human rights include:

    A right to life.

    A right to liberty and freedom.

    A right to the pursuit of happiness.

It's a matter of emphasis and style.

Both are correct.

For:

I have the right to _____

You are asserting/declaring that you do indeed possess a specific right.

In the case of:

I have a right to _____

You are more stating that you are justified to ____, that you are within your rights to do ____.

According to the Cambridge Dictionary

the

used to refer to things or people when only one exists at any one time

From the Wikipedia

Suffrage, political franchise, or simply franchise is the right to
vote in public, political elections (although the term is sometimes
used for any right to vote).1[3] In some languages, and
occasionally in English, the right to vote is called active suffrage,
as distinct from passive suffrage, which is the right to stand for
election.[4] The combination of active and passive suffrage is
sometimes called full suffrage

Suffrage is the right to vote in public political elections.

Usually we just have one right to vote described in the Constitution of our country.

So the correct sentence is

I have the right to vote, because I'm 18

But you could also say

I have a right, the right to vote.

A right is used to refer one element in a class of objects, one of your many rights; the right to vote is the only right, a specific right, that you have at that time that allows you to vote in public elections.

I'm not a native, but let me show a slightly different perspective.

In most cases you can look at rights from various perspectives, causing them to have various articles. The thing is those perspectives to some level interfere, making it actually correct to use both articles in more or less same cases.

Let's focus on your examples.

Human rights are inherent to every human. It means every human has their rights. As you can see you can then look at rights in general, so you use the undefined "a" article.

If you speak of a specific person and theirs right, you now restrict yourself more. This is theirs inherent right. So it uses now the specific "the" article.

Yet, you do not clarify, when you speak about human rights, if you refer to the right of a specific human or a general right of each human. So you can safely use either of the articles here.

In some cases you focus on a specific right, as it constitutes something. In such case the specific "the" article is required:

The right to live should be our consideration when we think of a life sentence

But for most other cases various views make it OK to use both articles in seemingly same examples (as those you provide).

For the case of voting rights this is pretty much similar. Your voting right can be generalised even further. Even if you look from a single person's perspective, you may consider it a generic right to vote, that is granted for all elections and referenda that might take place in your life, or the right to vote of a specific person.

So as other have already pointed out, in almost all cases both "a" and "the" article are interchangeable, however in some very specific cases one might be preferred over the other.