When a general goes to the graveyard in EDH, does his casting counter reset in the command zone?
You were correct in the assumption that the tracking of casting cost is not a counter, but a separate attribute of the game.

The official EDH rules state:


  While a Commander is in the command zone, it may be cast. As an additional cost to cast a Commander from the command zone, its owner must pay {2} for each time it was previously cast from the command zone. (ie: it costs 6WW to cast for the third time.)


The Magic comprehensive rules state:


  903.10. A player may cast a commander he or she owns from the command zone. Doing so costs that player an additional {2} for each previous time he or she cast that commander from the command zone that game.


Neither of these refer to "counters" at all - it's a static "every time you play this again, it costs {2} more."