playBunny: It would be different if the web page were to explicitly state that either or both dice may be used

Well, it does. At the bottom of the rules pages, under the heading Other important rules:

• If a player wants to use the second die first, he/she must click on "Swap dice" link below the game board. There are several situations when the link is not shown:
  • Both dice show the same value.
  • The player could not make a legal move with the second die.

And further up the page, when discussing movement of the pieces: If two different numbers are rolled, the player can make two moves at this turn. He/she takes the first (left-most) die value and moves one piece the same number of spaces. Then he/she does the same for the second die.

To me, there's no question about it. Moving off any die first, even if that blocks movement off the second die is allowed. If it were illegal, you wouldn't be able to swap the dice.

As Wil says, if he were playing on a physical board he would do the right thing.

I've never played backgammon on a physical board. Of all the games that you can play here, I've only played a few on a physical board, and backgammon isn't one of them. (Chess, Anti-Chess, Checkers, Halma, Line4, Reversi, Battleboats, Battleboats Plus, that's about it).