No part.
I realize that the word “doubt” appears in both phrases, but that does not mean they have the same meaning, especially since one is a legal term and the other is not.
Finding someone "guilty beyond a reasonable doubt " is not the same thing as “benefit of the doubt is applied to defendant.” as you can readily see:
give somebody the benefit of the doubt
to believe something good about someone, rather than something bad, when you have the possibility of doing either
give somebody/something the benefit of the doubt
to decide you will believe someone or something
benefit of the doubt
a judgment in one’s favor when the evidence is neither for one nor against one.
When giving someone the benefit of the doubt, you are believing what they say and taking their word because you, yourself, have some doubt about what happened.
benefit of the doubt
noun
a favorable opinion or judgment adopted despite uncertainty.
I repeat, since you seem to have missed it the first time:
There is no such requirement for the jury to give the defendant the benefit of the doubt at all. Zimmerman claims X - the jury has the sole responsibility of deciding how credible they find him, and decide that they do not believe him, even though they cannot be certain. If they had to give him the benefit of the doubt, as you would have it, they would have to accept that what he said is true. They would have to, even thought they find him to be a raging liar, accept an “inference favorable to the defendant”.
But no, they don’t have to, at all. They have complete freedom to think he’s a total liar and not believe a word he says, giving him zero benefit of the doubt.
Which is not the same thing as convicting him because the case against him has been proven beyond a reasonable doubt. They are separate and distinct things, even though they sound similar.