The fact that it's confusing should clue you in to the fact that needlessly gender-specific pronouns are treacherous: you're not even actively aware of the gender assignment you make at the first pronoun, but are instantly aware when the assignment is violated.
So my argument for her or him, he or she, rather than they, is that her or him, he or she, in documentation humanizes the documentation. It piques my interest. There is an active person here.
It's not boring, dead trees, produced for some deadline documentation, Sally is doing something with this code! Bob needs to stop sending data!
He or she are far better to see in code comments for the same reason it's fun to encounter latin, or star trek quotes, or even curses.
They doesn't have that effect on me. They tells me the documentation was done grudgingly, likely by a prig. It is likely formally correct and will still say nothing, or be completely obtuse and thick.
If you want to humanize the documentation, introduce actual characters. "Here's Bob. Bob wants to update his widgets." Don't pretend that stereo instructions can be made to read like Elmore Leonard stories simply by changing pronouns.
At the point where you start synthesizing vibrant life stories out of gendered pronouns with "the user" as antecedent, you might consider instead just conceding the argument.
