I'm currently trying to get off my 5'10 Air rush directional. I want to go to a bi-directional/twintip/wakeboard style kiteboard. Can someone outline the differences between the three for me??

I just want a board that I dont have to gibe on and I dont want boots on my board, I like my foot straps.

Oh yeah I'm 65kg and I fly a 7m Warrior and a 15.5m Titan. I would like to pay less than $500 for a second hand board.
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White men can jump! (With the help of my 7m warrior.)

Bidirectional and twin tip are interchangeable terms, you can go in both directions equally well because the shape of the board at both ends is identical. At first glance people can sometimes not discern between a TT and a wakestyle board. In shape outline they are often pretty similar. Some general rules though are a wakestyle board is often quite small, they have significantly less volume and off course use boots. The small size and the low volume combined with the weight of the bindings make the board heavy; you have to be powered to ride your wakestyle board. TT's usually have more volume and can get quite big in length and width hence you more surface area and better bottom end and ease of learning for beginners. Additional but important things like concaves, fin foil, size, and placing, rail shape etc all influence how a board will ride.

Real wakeboards used behind boats often have heaps of rocker, are very wide and heavier than kite wakestyle boards. These all combine to make the board way too power hungry for versatile and satisfying use with a kite. Looking at a TT at your bodyweight and considering your kite sizes have you covered for a wide windrange I think a board between 125 - 145 x 35 - 42 would be a good all-rounder. I'm sure others will have opinions on size.

All the best,