1) The numbers that keep changing before you hit attack do matter. (I believe thats what numbers you are talking about). The better your soldiering and tech as well as the invcasion tactic you chose will determine what your numbers are verses what the AI has.  WHen you hit attack, the numbers freeze, they will determine how many soldiers you lose and what the AI loses.

So you will try to get a much higher number than the AI.

2) The screen for plantetary invasions was changed per player feedback. I don't know whent the tutorial shows but its probably the old screen (still current on DL and DA) where you get a bunch of tiny dudee and boxes fighting it out. If you see something different its just because the devs made up a new screen some time ago.

So it is a good idea to look at the numbers and try and press the button when you have a high one - a bit like rolling the dice in a board game, there is the basic mods for the battle and the flashing numbers are the random bit.

IIRC the soldiering advantage (and choice of invasion tactic) is what is used to determine what the range of values are available when rolling the attack advantage.