The <PERSON> brothers (now sisters) got <PERSON> to do the director's commentary on the films - so not quite from the horse's mouth, but close. The 'meaning' of the trilogy (of course it has many meanings) is discussed here, as follows:
Thus, as <PERSON> summarizes a more integral interpretation (that takes into account what is revealed in all three films), Zion represents body (filmed in blue tint), the Matrix represents mind (green tint), and the machines—this is the kicker revealed in part 3—represent spirit (golden tint). For those of you keeping track, this is indeed quite similar to the Great Nest of Being as taught by the world’s wisdom traditions, a spectrum of being and consciousness reaching from body to mind to spirit.
Borrowing from the wisdom of Christian mysticism, “The flames of Hell are but God’s love denied,” and so an alienated and dissociated spirit manifests as an army of machines bent on destroying humankind. It is only in the integration of body, mind, and spirit that all three are redeemed and peace returns.
I think the numerous allusions to the number three all over the trilogy are mainly a reference to this. The human mind, body, and spirit are at war with one another. Each of them is crippled in isolation - the body without the mind is empty, unaware, sleep-walking through life, used as no more than a source of power like a battery. Living the life of the mind without the body is a miserable existence, no bodily pleasures. The spiritual life is at war with both the corruption of the body, for which it has bottomless contempt, and the moral rebellion of the secular mind, in which it sees its own destruction. But the war cannot be ended by any one triumphing over and destroying the other two, but only by making peace between them, tolerating differences, letting people make their own choices.
It's also worth considering <PERSON> referring to the films as an allegory for being transgender (mind and body at war...). There are subtle references to the binary, and the attempts to escape from it. Ther word 'Cypher' originally meant zero, 'Neo' of course is 'The One', 'Switch' might be a reference to to something that can be in two states, a binary bit, (the character of <PERSON> was originally meant to be explicitly trans, female inside the matrix and male outside,) and 'Trinity' goes beyond the binary. 80
You don't have to accept the binary this-or-that, red-pill-or-blue-pill, male-or-female, yin-or-yang options offered to you, you can take a third way. It's all about choice.
<PERSON> : My name's <PERSON>.
<PERSON> : The <PERSON>? Who cracked the IRS d-base?
<PERSON> : That was a long time ago.
<PERSON> : <PERSON>...
<PERSON> : What?
<PERSON> : I just thought... you were a guy.
<PERSON> : Most guys do.
A female hacker would probably have a thing about not fitting in to the conventional, binary choices on offer from society. <PERSON> is very much not a girly-girl - tough, gun-toting, competent at riding motorbikes and flying helicopters, short hair... And that's something that fits with the convention-breaking non-binary sex-role theme, as well. Online, you see her real personality, her mind rather than her body. So it may be that her choice of hacker name was about going outside and beyond the binary choice. That's just my own speculation, though.
The <PERSON> have said that all the character names have multiple meanings, so there is almost certainly more to it, and they have deliberately chosen not to talk about it so as to leave room for people to find their own interpretations. There are lots of things that come in threes - it's a common trope in stories. Three witches, three wise men, three bears, ... There's something satisfying about the number. So there are lots of things it might refer to. 861