Decided to write a new post so people don't have to go back and look, sorry for the previous one, was falling asleep while trying to write it!
Something interesting to note is that this was shown being played with a Dualshock 4, I don't understand Japanese enough to translate the dialog, but suffice to say, this was running on a PS4.
Names read:
ノクト (Noct)
グラディオ (Gladio)
イグニス (Ignis)
プロンプト (Prompt0)
Both Noctis and Gladiolus are using their nicknames in the battle screen, very cool!
One thing to point out now, it seems that each row of the command menu on the left corresponds with the character across from it, this may change when you move into a menu, but initially this is the default state.
One of the first things we see is this:
ワープ (Warp)
As shown in the video there are fixed points in which Noctis can warp to. The command for Warp will not always show up, it will only be displayed when Noctis can see an available warp point.
Once the enemies appear, Noct assumes an alerted state and the first option changes to:
ぼうぎょ(Defence)
with the option itself actually highlighting and expanding, as does an option for Prompto and soon after other party members:
こうげき (Attack)
Each three of the other party members are showing "Attack" with arrows pointing at ゴブリン (Goblin).
Note that Gladio, Ignis and Prompto start off in the "Defence" state, and switch to the "Attack" and we see they are attacking various Goblins in the area, sometimes switching back to the "Defence" state.
And of course, Gladiolus starts attacking ベヒーモス (Behemoth) as do some of the other party members.
Then, Noctis himself is seen attacking as well with the Goblins in the area, I noted that before he does start attacking while in the Defence state, he seems to dodge and try to avoid enemy contact automatically perhaps?
Then, Noctis wants to attack the Behemoth himself, and we see ファイア (Fire) being used against the Behemoth.
There doesn't seem to be any special indication here when you switch to Ignis for a second with his fire sword attack, so I'm not entirely sure what's going on here...
Next after a little more combat we see Noctis warp up and the command to warp attack becomes available as ワープ攻撃 (Warp Attack) the [攻撃] itself is the same [こうげき] as before, but the kanji instead of hiragana, Noctis selects the Goblin below and warps into it attacking.
Nothing else is shown that we haven't seen already dialog wise, warp attacking, a Fire at the end.
No clear indication is shown as to when you will attack with allies, or anything it seems, perhaps something you can trigger that we just don't know yet, or perhaps it was explained in the voice over that I'm unable to translate for you.
I hope this serves some help to understanding some of the menu system and text shown in this demonstration!