When you play the Dragon for the first time, you are supposed to place the Dragon on the start tile. By 'played' does it mean "placed on the table" or "character moved onto"?

You place her when the tile is first played into any unoccupied location.

I think what you're asking is whether the dragon is first "played" as a result of a tile being laid or a hero moving onto the "Move Drakon" tile.  You can place the dragon figure on any unoccupied chamber the first time a hero STEPS ONTO the "move Drakon" icon.   

Actually, re-reading the rulesheet, I see what you mean.  It does seem to say that you can place the dragon when you lay the first 'move Drakon' tile.  We always play that she only gets placed when someone actually steps onto the tile....anyone have any thoughts about which version of the rule might work better, and why?  Anecdotally, it seems to work fine the way we play, but I'd be interested in hearing from anyone whose opinion differs. 

Zozimus said:

We played a couple of games placing Drakon only when a Drakon Moves chamber is entered by a hero. The rules seem to state otherwise: Drakon is placed when the chamber is placed.

This gives a little edge to the player who first places such a chamber.  He can choose where to put Drakon and therefore create problems to other players. Also entering a chamber with Drakon inside causes you to lose Gold and to be teleported back to start.

I think following the rules makes gameplay more interesting and I recommend it.