I'm sure they fought with both styles but I'm still curious about the major differences in portrayal.
This is what I see whenever I look at ancient Greek art and pottery: http://www.timetrips.co.uk/gw-hoplites-fighting.jpg
And this is how games/movies/modern art portrays them: http://cloud-4.steampowered.com/ugc/531747978942698104/FCD23DDD24784347CF41CBC88E5BF779CFA2192C/
Am I missing art and military memoirs where they are described to fight with the latter form? Was it just for the sake of the art style that they are depicted holding them above their heads? Or does pop culture mis-portray what a true Hoplite unit's formation would look like?
I am not an expert in any fashion on this topic but I'd like to provide my thoughts on the subject. In the type of hand to hand combat that the Greek pottery depicts above, you see line fighting, which in this case includes a shield wall. A tight wall would limit the amount of space available for utilizing a spear from lower angles as well as force the combatant to change the angle of the body, which would affect the wall itself. Top down fighting in spear tactics allows individuals to still be active in shield wall fighting while "holding the line" [this large article here] (http://www.swordsmanship.ca/academy-articles/fighting-with-staff-and-spear/) discusses proper staff and spear techniques that would have been historically used in actual combat that we don't see today.
You fight to the openings that are presented, which means, there is no one way to start and finish a fight with staff (or any other weapon for that matter), only what you see your enemy leave open to you in their stance or attack and how you choose to exploit it.
If you are fighting body armor and another shield wall, utilizing your spear from the side is not only potentially ineffective, but also dangerous to your health and the health of those you are defending.
Where the "video game" mentality comes from in my opinion is the bayonet style fighting of the past two world wars. A downward thrust is much more easy to defend than an upward thrust (which you can see in your video game link. [This link here] (http://www.globalsecurity.org/military/library/policy/army/fm/3-25-150/ch7.htm) show's proper use of a bayonet in the military and basic military training has referred to the difficulty of blocking an upward thrust by an unarmored combatant.
The use of the downward spear is due to tactics and formation in Greek depictions. Video games that depict a side spear are not due to historical accuracy, but glamorization of current trends.
As a follow up, what about in the later Macedonian sarissa phalanx? Was the under-hand grip prevalent (like in the movie Alexander)? Surely the longer spears couldn't be used overhand...