It's not well-worded to be fair.
But in general the person operating the barriers can set them to reject whatever tickets they want, and I assume that they can choose whether or not to retain/return tickets too.
Generally the person operating the gates can do the following:
a) No Geography - checks date/time, but DOESN'T check origin/destination so gives you ticket back because it isn't sure whether it is a complete journey.
b) Full Checking - checks everything, retains ticket when completed or returns for you to carry on if not.
c) Accept All - checks that it is some sort of "rail" ticket but nothing more. Always returns to you.
d) Special Rules - allows you to reject up to 20 different NLCs, allows you to reject specific ticket types or specific passenger status codes, (railcards, LACON etc).
You can also choose from pre-programmed options with regards to CHILD checking: Monitor, will flash orange but return ticket. Alarm, will beep but allow you to enter/exit. Reject, Will flash, beep and reject ticket.