hirschjt you remembered correctly but it's too late now.
When you arbitrate a player you are shown the results screen. That's the player's contract for this season. If you use the Long Term Contract link immediately on that page only, you will be able to negotiate a LT deal that goes into effect the next season.
If you leave the results page at all, that link/option is gone for the season. You will still be able to LT the player before his 3rd arb next season.