Is intention the result of rational or logical thinking or some other?

When you think rationally, you may not have the intention to harm others. But in a moment of anger when you are not thinking rationally, you might suddenly have the intention followed by the action to cause harm.

Harm caused by anger to one who is angry is described in the Kodhana Sutta, that I quote here:

An angry person is ugly & sleeps poorly.
Gaining a profit, he turns it into a loss,
having done damage with word & deed.
A person overwhelmed with anger
    destroys his wealth.
Maddened with anger,
    he destroys his status.
Relatives, friends, & colleagues avoid him.
    Anger brings loss.
    Anger inflames the mind.
He doesn't realize
that his danger is born from within.
    An angry person doesn't know his own benefit.
    An angry person doesn't see the Dhamma.
A man conquered by anger is in a mass of darkness.
He takes pleasure in bad deeds as if they were good,
but later, when his anger is gone,
he suffers as if burned with fire.
He is spoiled, blotted out,
like fire enveloped in smoke.

When anger spreads,
when a man becomes angry,
he has no shame, no fear of evil,
is not respectful in speech.
For a person overcome with anger,
nothing gives light.

"According to one of many sources, the definition of intention is described as 'Right intention [...]"

Notice that the source you link to is not defining "intention" (aka "volition" from pali cetanā, or the related "volitional formations", "kamma formations" or "formations" from pali saṅkhāra), the psychological aspect of the mind associated to kamma, but "Right Intention" (also translated as "right thought", "right aspiration", "right resolve" from pali sammā sankappa) which describes instructions to develop oneself in the path, not an element of the psychological framework.

"If my thought is the result of anger however i do not have intent to harm is that not the result of rational thought?"

Just to contextualize:

[The Buddha:] "What do you think, Rahula: What is a mirror for?"

[Rahula:] "For reflection, sir."

[The Buddha:] "In the same way, Rahula, bodily acts, verbal acts, and mental acts are to be done with repeated reflection.

"Whenever you want to perform a bodily act, you should reflect on it: 'This bodily act I want to perform — would it lead to self-affliction, to the affliction of others, or to both? Is it an unskillful bodily act, with painful consequences, painful results?' [...]

"If, on reflection, you know that it is leading to self-affliction, to affliction of others, or both... you should give it up. But if on reflection you know that it is not... you may continue with it."

-- MN 61

If we understand a thought as an (volitional) act of angry thinking with intention to harm, and upon seeing the danger in such thinking, one abandons the intention to harm, you could call this reflection a "rational thought". Afterwards, while still angry, if one has another thought, a non-harming thought, this is another act of thinking with another intention behind it.

I guess you could say this second intention is the "result of a rational thought", but only to the extent you're explaning that this previous reflection influenced it.

But to the extent that "result" means a much closer relationship, I think it would risk hiding important details:

"Intention [cetanā], I tell you, is kamma. Intending, one does kamma by way of body, speech, & intellect.

"And what is the cause by which kamma comes into play? Contact is the cause by which kamma comes into play.

-- AN 6.63

"And what, bhikkhus, is dependent origination? With ignorance as condition, volitional formations [saṅkhārā] come to be; [...]

-- SN 12.2

"If i do act irrationally, to save or protect a life, is that not intention without thought?"

If one acts "irrationally to save a life", I'd say he is acting conscious and deliberately, not irrationally: that's his intention after all, "to save" (*). If he saves "by accident", then I'd say it was an unreflected, unaware act (with regards to that specific outcome, at least), with no intention to save, but with intention to, say, obtain something else (which, by accident, resulted in saving).

From my understanding of Theravada point of view, (and grossly speaking) intention is always behind an (bodily, verbal, mental) act. Intentions, however, are subject to be reshaped by one's own will.

Finally, intentions may be succeeded by careful reflection, or not. They may be shaped by previous introspection, just by deep habits, etc. or any combination of it.

_{(*) He may be "irrational" in his method, not in absense of intention or thought.}

Dependant origination(DO): depending on contact feeling arise, depending on feeling craving arise, ..craving..clinging, ..clinging..becoming, ..becoming.. there is birth. etc..

From contact until the point of birth there is no kamma (to be precise it's becoming).

Birth of what?

Birth of another thought, of a person committing an action good or bad.

Feeling of anger leads to birth of angry thoughts. If the thought is hold onto and contact (bringing to mind of that thought), then feelings arise again( it may again be anger or sympathy depending on the thought) and the whole chain of DO repeats.

Through mindfulness of feeling we let it ceased before it goes to craving(or aversion).

When there is craving we let it go before it becomes clinging. When there is clinging we let it go before it progresses to becoming. When becoming, we let it go before the birth of another thought or action.

Intention comes after becoming and when mindfulness is strong there is some space to reflect before the birth of acting. That's how I understand DO to operate.