You're right--people should not project their feelings onto others. People should work to distinguish between what was actually said and what was understood.
By my understanding, Kayla feels like you were being nasty towards her, she feels like you never talk to her "that way", she feels like you are "slapping her down".
Let's take a look at what you said that rubbed her the wrong way:
I'm gonna go out on a limb and guess that it was the first line that rubbed her the wrong way. In a sense, you are projecting your feelings about her onto her ("You are being awfully defensive").
Kayla responded very respectfully at first:
Here -- without projecting any feelings on you.
Your next response is where the whole thing goes south:
Here -- where you make statements such as, "take responsibility for what you do", "stop blaming me", "stop projecting your thoughts onto me, pretending I said or implied things I did not"
But, she had not yet blamed you are accused you of anything. All she said was: Your picture has no correlation to what she was saying.
By my understanding, your need to "be right" trumps compassion and empathy, in this instance.