Definitely a later date.
Also, I feel like I here both back and out pretty regularly, and up most commonly as the opposite.
Honestly, I never have heard that!
Yup I’ve heard that.
I grew up in Saskatchewan where they don’t use DST and I’m still confused by “spring ahead” and “fall back” when referring to time changes. What is going ahead – the clock or me? Or are those both the same thing?
A later date.
And there’s really no reason to use “back,” “out,” “up,” etc. Simply say, for example, “The event was on March 5th but we moved it to March 12th."
Ha! The “spring forward, fall back” expression is a great illustration of what I’m talking about! In the fall, we (outside of Saskatchewan! 
) move the clocks back to one hour earlier. Makes perfect sense – the clock moves clockwise indicating the flow of time, and in the fall we move it back – counter to that direction – by one hour. 2 AM becomes 1 AM. So how the hell is moving an event back in time supposed to mean changing it to a later time or date?
The same with calendars and timelines in general. If there was a wall calendar with a bake sale on Wednesday and someone said, “move the bake sale back two days”, the most obvious thing would be to move the bake sale backwards two days and put it on Monday. And time machines. If I ask my assistant move the destination date two days back, my intention would be to go into the past by two days. Won’t I be surprised when I end up two days in the future!
The English language seems to be ill-equipped to coherently communicate relative time. The one that drives me nuts is “this”, “next”, and “last” in relation to days of the week. Say it’s Tuesday. Is “last Monday” yesterday or 8 days ago. Is “next Monday” in 6 days or 13? But apparently, if you mean the actual next Monday, you need to say “this Monday”.
It’s fuzzy. It more seems to be how far you are from the day. If it’s Tuesday, “last Monday” would be 8 days ago. But if it’s Saturday, “last Monday” would be 5 days ago.
“This” is always the very next one coming up.
Yes, this is another one that bugs me. And there’s no definitive reference manual for it because different speakers actually do mean different things by those expressions. Sometimes the meaning is perfectly obvious – if you’re in a regular Monday morning meeting, and someone says “we’ll discuss this next week” it’s obvious that it means the next meeting a week from now. But yes, in a great many cases the meaning is extremely ambiguous. Just say what you mean, dammit – give dates and times, stop referring to moving things back, forward, or sideways!
I have the same problem with bi-weekly meetings. Is it twice a week or every other week?