with power applied, touch BLUE-to-YELLOW. you should hear motor whirring, lightings start flashing, gate arms swings down. arm motion will automatically stop by itself when reaching bottom - takes 2 or 3 seconds for the arm to go down. if you disconnect BLUE-to-YELLOW during motion, it will stop moving.
with power applied, touch BLUE-to-WHITE. you should hear motor whirring, lights are already flashing, gate arm swings up. arm motion will automatically stop when reaching top - also takes 2 or 3 seconds. lights stops flashing when arm reaches top. if you disconnect BLUE-to-YELLOW during motion, it will stop moving.
so, with that background:
A so-called SPDT toggle switch (3 terminals) connected as shown will do the trick. One position is down, other position is up. As mentioned earlier the arm motion automatically shuts off when reaching end of travel so it's OK to use a toggle switch.
But, you mentioned pushbutton switch. You'd need two momentary type (2 terminals each, spring-loaded to turn off when released). Push one button for down, push other button for up. To complete the arm motion in either direction you'd have to hold the button until the arm reaches the end of travel.
Irrespective of which configuration, the current flowing in the switch is very small (much less than 1 Amp) so essentially any switch you can find will work.
As shown you don't need to hook up the BLUE wire on the 2nd gate if operating them in tandem.