What is temptation?

Temptation is one of those words that shows just how.... difficult English can be.

I am assuming that when you talk about the temptations and Christ, you are talking about the temptations that were apart of his 40 day fast and the ones immediately following that fast?

Now the scripture in Matthew reads:

Matt 4: 1,3 ) 1. Then was Jesus led up of the Spirit into the wilderness to be *tempted [sic] of the devil. 3. And when the tempter came to him he said, If thou be the Son of God, command that these stones be made bread.

So we have Jesus going up to be tempted, and we have the tempter coming to Him once he is done.

Now, as noted in your definitions, Temptation can be applied both as a adjective - when it describes a condition one is experiencing - and as a verb - when it describes an action. No where in these verses, does it apply temptation as an adjective that Christ is experiencing. In both verses it is being used in verbal form. First He is going out to be tempted, (be being key here to the verbal application of the word).

Then He is confronted with the tempter, again the verbal form of the word (though it can be confusing because here the verbal form is being used as a pronoun).

If you keep reading and come to Christ's responses, it is fairly evident that tempted is not a good adjective for how He feels when presented with the devil's offers.

Matt 4: 4,7,10) 4.But he answered and said, It is written, Man shall not live by bread alone, but by every word that proceedeth out of the mouth of God. 7. Jesus said unto him, It is written again, Thou shalt not tempt the Lord thy God. 10. Then saith Jesus unto him, Get thee hence, Satan: for it is written, Thou shalt worship the Lord thy God, and him only shalt thou serve.

None of Christ's answers show any desire to partake of the temptations, or that they provoke any sense of allure in the Savior. In fact He rather forcefully rejects the offers and then casts the tempter out of His presence.

So TL:DR While Christ is presented (verbal form) with temptations, He never feels (adjectival form) the temptations. Hence He does not have a sinful nature.

*As a side note, in verse 1 of my KJV of the bible the footnotes offer 'tried' or 'tested' as an alternate translation for 'tempted'. This seems to flow better with the fact that the devil shows up only after Christ comes back from the wilderness. (Again, that's just a note and my opinion on the whole scripture, and has nothing to do with my answer!)

The following is sort of an answer to your question: the desert fathers (from the beginnings of monasticism) thought a lot about the beginning of sin.

Sin has its origins in "logismoi," thoughts in our mind floating hither and thither, often from whence we know not. "Logismoi" is usually translated as "thoughts," though maybe in this usage it would be more suggestive to translate it as "distractions". Having logismoi is like walking the streets of Las Vegas, where promoters snap packets of prostitution flyers loudly across their hands and extend the flyers toward pedestrians. It is no sin to be assaulted by such advertising.

One may (at one's own risk) enter into dialogue with the logismoi to see what holds them together and to see how to help your brother (and yourself) tear them apart when need be.

Sin begins when one "couples" with logismoi, a condition described in depressing detail in our record of the ancestral sin:

... the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was to be desired to make one wise.... (Genesis 3:6)

One might say sin begins when one seriously considers a wrong choice as an option. Our Lord spoke of this in His sermon on the mount:

You have heard that it was said, "You shall not commit adultery." But I say to you that every one who looks at a woman lustfully has already committed adultery with her in his heart. (Matthew 5:27-28)

I say this is sort of an answer because it doesn't address the use of the word "temptation" in English as such. But I hope it helps illuminate how sin begins.

In particular, Christ was, presumably, assaulted by logismoi as much as, and probably worse than, anyone else in human history. But He never coupled with them, so He never sinned. E.g. when tempted in the desert, Christ made it clear with His curt replies to Satan how little He thought of Satan's advances.

More terrifying to consider along these lines was His struggle in Gethsemane. Perhaps it's best for me not to speculate on what was going through His mind then.

I think @ryan hit on the desert fathers' distinction. The logismoi are the "presentation" of temptations, and the coupling is the "feeling" of temptation.