The issue of whether the automatic addition of 30 minutes should be triggered by the expiration of the initial control, or by the move count, is a hot-button issue that has been debated at length on these forums.
If the clock is set to count moves, and to add time when move 40 has been reached, then white’s wish (in your original example) for documentation that black has made the time control, is satisfied. It’s less confusing that way.
On the other hand, there is always the possibility that the move count is off. One of the players may have forgotten to press his clock a time or two, in which case the move count will be too low. Or, perhaps an illegal move was made, and the clock pressed, when the opponent announced “Illegal!” and pressed the clock back. In this case the move count will be too high. Or, the game may have started by white playing his first move and then starting black’s clock, rather than by black pressing white’s clock to begin with. In this case the move count will be half a move too low.
TDs have strong opinions, on both sides, about which setting is preferable.
I think it’s unwise, though, for a TD to try to impose his will (either way) on the players. Let the players choose for themselves. If there is an argument, let the player who furnished the clock decide the issue.
Of course, if the organizer furnishes all clocks, then the organizer can set them the way he wants to.
Bill Smythe
I impose my preferred settings on the clock since the players do not know how to set them.