The first response here is that you can't force the first movie to play by the second movie's rules. If someone broke continuity here, it's the second movie. There's no way the second movie can be considered consistent yet the first movie must have gotten it wrong - that's the wrong way round.
But even if you assume that both movies use the same rules for resolving the game using time travel, the basis for your question is slightly wrong.
So, this is a bit complicated to explain. The short answer here is that the kids you see at the Christmas party are younger than the Judy and Peter who move into the Parrish manor in the beginning of the movie. Therefore, the kids at the Christmas party will never "remember" the game even if their consciousness jumped back to the point in time when they started playing the game (which is what happens for Alan and Sarah).
A simpler way to think about it is that Alan and Sarah's consciousness jumps back to 1969, the exact time where they started playing the game.
So if we were to see a scene that takes place in 1968, it makes sense that 1968 Alan and 1968 Sarah wouldn't seem to remember anything about the game being played.
But if we were to see a scene that takes place in 1970, it makes sense that 1970 Alan and 1970 Sarah would remember anything about the game being played.
The logic here is simple: the time you start playing the game (let's call it T) is the exact time you jump back to when finishing the game. Therefore, the "you" that exists after T will remember the game, but the "you" that exists before T will never have those memories.
So let's set T to the point in time where Peter and Judy started playing the game.
The Christmas party scene where Alan and Sarah see Peter and Judy takes place before T. This Judy and Peter (at the party) are younger than the Judy and Peter who move into the Parrish manor with their aunt and start playing the game. Their parents specifically mention that they are intending to go on a skiing trip, whereas the Judy and Peter we meet at the beginning of the movie were orphaned after their parents already died on said skiing trip.
Let's forget all these branching timelines, and see what things must look like from the POV of a third observer:
- Alan Parrish goes to his house with Sarah.
- Alan and Sarah inspect this boardgame that Alan found.
- Out of nowhere, they hug and this scene plays out.
- From then on, they will live their life with the memories of the Alan and Sarah who finished the game and jumped back in time to 1969.
- Lots of things that happened originally won't happen, like Alan's dad ruining his company because he's looking for his son etc.
It's not as easy to do this for Peter and Judy, because pretty much everything we know about them (being orphans and moving with their aunt) is going to irrevocably change. What we can infer, however, is that at some point in time, out of nowhere, Peter and Judy's consciousnesses land, and from then on they will live their life with the memories of the Peter and Judy who finished the game and jumped back in time (which for them is only about a day back in time, but their life will be dramatically different due to timeline changes).