Double moves are not only related to war, but in this specific case it is, since a war declaration would appear to be imminent (also supported by the "...yet" statement you give, HBHR). Any movement of a unit that causes one player to gain an advantage over another player, by not giving the other player time to react to the first move, is a double move - even at peace time
An example of this is if two players are in a settling race for a nice city spot. Consider an example where both players have their settlers placed in such a manner that a city can be built by both of them in the specific spot this current turn. Player 2, however, moved his settler into the spot he's now standing after Player 1 moved in the previous turn (so the turn order was Player 1 -> Player 2). If Player 2 plays before Player 1 in the turn order in the current turn, and thus settles the city, he has effectively double moved Player 1.
When considering whether your move is a double move or not, always consider how the move would affect the other player(s), and whether the move performed would give you an advantage in a game that has simultaneous turns as opposed to a game with non-simultaneous turns. If you see that doing a move does indeed give you an advantage, because of the possibility to move units twice in a row in a simultaneous turn environment, then don't perform the move until after the other player(s) have been given the opportunity to react to the first move before doing the second.
As this is a first time situation, I don't think any penalty is needed. A reload to the start of the turn should be sufficient to resolve this. I am not doing a reload just yet though, as I will give any player who is interested in making comments time to do so first.