Why did the protection spell work for Harry?

I'm new here(actually posting at least) so forgive me if I'm doing this wrong.

I am a little confused as to why Harry's sacrifice in the forest gave the same protection to the people he loved as what he got from his mother.(If that is indeed what happened?)

The way I understand it his father did not give him and his mother the protection as he was not given a choice about dying or not. His mother did- therefore she gave him the protection.

JKR: James was immensely brave. But the caliber of Lily's bravery was, I think in this instance, higher because she could have saved herself. Now any mother, any normal mother would have done what Lily did. So in that sense her courage too was of an animal quality but she was given time to choose. James wasn't. It's like an intruder entering your house, isn't it? You would instinctively rush them. But if in cold blood you were told, "Get out of the way," you know, what would you do? I mean, I don't think any mother would stand aside from their child. But does that answer it? She did very consciously lay down her life. She had a clear choice -

ES: And James didn't.

JKR: Did he clearly die to try and protect Harry specifically given a clear choice? No. It's a subtle distinction and there's slightly more to it than that but that's most of the answer.

Now the way I understand it Harry did not have the choice his mother had. Yes he could have fled Hogwarts and never go into the forest to Voldemort- but then the same could be said for his father as he could have escaped the building through a window/backdoor instead of charging at Voldemort in the corridor.(Yes I know, neither Harry nor his father would have done that, but that's beside the point).

Voldemort was definitely going to kill Harry in the forest, just like he was definitely going to kill James. So my question is:

If Voldemort intended to kill Harry, why did the protection work for the people he loved?

Canon answers please (books or JKR-herself-recognized websites/forums, no wiki/wikia please!).

Answer

Intent is the difference i think.

James intended to fight Voldemort

from deathly hollows chapter 17 Bathilda's secret

He was over the threshold as James came sprinting into the hall. It was easy, too easy, he had not even picked up his wand. . . . “Lily, take Harry and go! It’s him! Go! Run! I’ll hold him off!” Hold him off, without a wand in his hand! . . . He laughed before casting the curse. . . . “Avada Kedavra!”

Lily never intended to fight - she intended to sacrifice herself hoping against hope Harry would be spared (Voldemort is a cold blooded killer after all).

from deathly hollows chapter 17 Bathilda's secret

He forced the door open, cast aside the chair and boxes hastily piled against it with one lazy wave of his wand . . . and there she stood, the child in her arms. At the sight of him, she dropped her son into the crib behind her and threw her arms wide, as if this would help, as if in shielding him from sight she hoped to be chosen instead. . . . “Not Harry, not Harry, please not Harry!” “Stand aside, you silly girl . . . stand aside, now.” “Not Harry, please no, take me, kill me instead —” “This is my last warning —” “Not Harry! Please . . . have mercy . . . have mercy. . . . Not Harry! Not Harry! Please — I’ll do anything —” “Stand aside. Stand aside, girl!”

Harry never intended to fight. Harry intended to sacrifice himself hoping against hope his friends would be spared (Voldemort is a cold blooded killer after all).

from Deathly Hollows chapter 33

Harry understood at last that he was not supposed to survive. His job was to walk calmly into Death’s welcoming arms. ....

...If he could only have died like Hedwig, so quickly he would not have known it had happened! Or if he could have launched himself in front of a wand to save someone he loved. . . . He envied even his parents’ deaths now. This cold-blooded walk to his own destruction would require a different kind of bravery.

oh and since Love is also part of the protection. i'm going to say intent plus love.

ps. i'm not even sure "intent" is the right word to use. Since i have to account for what the characters were feeling while they were dying. "Intent" seems so inadequate.