The "Pe gathers, Pi organizes" is some mantra that Turi (in accordance with DaveSuperPowers of course) has been trying to promote, which further adds confusion to what things such as Ne and Ni actually mean. If you're to read Jung, you'd probably come away with the idea that it's Perception in general that gathers, and Judgment in general that organizes, and since he is the source of this terminology I don't get the point of redefining these concepts.
Time moves forwards, theories evolve, what's old is refined and/or expanded upon, and the new assumes its position - I don't recall it precisely, but Jung himself said something along the lines of the words of our fathers should be viewed as guidance, though not something to
attach ourselves to and strictly adhere to.
Maybe next time you talk to DSP you should tell him to change his terminology. If you're going to try to reduce things down to a few words, it helps to actually choose words that convey what you are trying to get across.
But even so, DSP's theory seems to be predicated on the idea that Pi is the opposite of Pe, and thus Pi's definition must follow from being the opposite of Pe, which frankly seems silly, as they are both perceiving functions. He does a similar thing with the judging functions too.
The definition of 'organize' makes perfect sense, though the people that are corrupted by internet terminology within the realm of personality theory will fight tooth and nail against it, as they are interpreting the word differently.
Here is the Oxford Dictionary definition of 'organize':
arrange systematically; order.
This is the idea that is being conveyed, it is the perfect word to use - and to people who aren't already poisoned, will make a brilliant kind of sense.
Si, and Ni, arrange systematically, the information they are receiving - and pass it off to the decider/judging functions.
It's literally that simple - in and of themselves, Ni and Si don't 'do' anything, they simply perceive - but it's "what" they perceive, which is where the organization aspect comes in.
It is not 'organize' in the sense of, turning all of the tins of soup in your pantry around so they all face forwards, it is not 'organize' in the sense of getting people together for a common cause, and ensuring everyone arrives at the destination at precisely the right time.
It is organizing the information it observes.
Si, with regards to 'what is' and Ni, with regards to 'what isn't'.
It's also true to Jung, read up on Jungs subjective impressions in the introverted sensation descriptor - the Si type is interested, in only what piques their interest - everything else, is essentially non-existent, Si is portrayed as Switzerland in some respects - this is 'organization' of sensory information.
When something piques the Si types interest, and provokes that sensory impression - then that information is being organized, arranged, and allowed through the neutrality filter - in order - the information is organized.
Say the Si type looks at a woman, the order in which the information comes in, will be in order of interest - bright red hat, big fur coat, leather boots - this is Si, organizing sensory data - it's in order of what has piqued ones interest.
Hopefully this helps.