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First of all, the divisions do not need to use the same names as 1993 and before; I was even thinking of possibly Howe and Richard Conferences with Hull, Lemieux, Orr, and Gretzky Divisions (I know Gretzky may seem like a better conference name but this gets the whole Mr. Hockey-Rocket Richard debate going).
Now that that's out of the way, here's my idea for scheduling.
Since two divisions have eight teams and two divisions have seven teams, the intra- and inter-conference play will not be equal for every team.
For the eight-team divisions, each team should play the other seven teams in their division six times (three at home three on the road) for a total of 42 games. They should play everyone in the other division within the conference three times, alternating each year who has two games at home and who has two games on the road.
This is a total of 21 games, making a total of intra-conference games 63. This would leave 19 games remaining for inter-conference play (assuming an 82-game schedule), which can be solved by playing each inter-conference team at least once and a random four of them from the opposite eight-team division twice, just like how each team plays three random inter-conference foes twice currently.
For seven-team divisions, each team should play the other six in the division six times each for a total of 36 games. They would obviously play the eight other intra-conference teams three times each, resulting in 24 of these games and 60 total intra-conference games. This leaves 22 games remaining on the schedule for inter-conference play.
Obviously, they should play each team in the opposite conference at least once while playing a random seven teams twice, making it only sensible to play all of the teams from the opposite seven team division twice every year to make it easier for the schedulers.
The playoff format would be the same as it is currently, but with only the top two seeds being division winners, since there are obviously only two divisions in each conference.