Type - This is a representation of your cognitive wiring (to use my father's analogy). It represents how your brain is wired up. I always saw this as each brain having the same wiring but there being less resistance and voltage drop over certain wires from birth and hence some favoured kinds of thinking appear. Of course the other wires can be upgraded or you can grin and bear the strain of using them every once in a while but that's never the route of least resistance.
Personality itself is too complex to be expressed in such simple terms as sixteen types with any accuracy at all and that's not really what the MBTI is about, though it often masquerades as such. Basically with the MBTI you observe the personality and, with the type, can decipher some of the more inobvious parts of that persons psyche.
For example what is the personality of an INTP?
I certainly don't have it and neither do you. What is this personality? We can certainly say what some of the features should be but only in terms of how thinks are processed and not what is included within that process nor can we say with any certainty what result the process will produce.
For example take intuitives. They link what they are thinking about to a resource of previous experiences and seemingly matching patterns. Well there's nothing about what they are linking to in specific and no note made about what result you would get (for one any attempt would produce a library of possibilities), as such how can it describe the personality of that person?
To put it in your words, is a leaf a tree? No a leaf is part of a tree but the tree is much, much more than it's leaves.