I am referring to this part (0:13-0:41), also known as the part 1 motif.
Now listen to this part (1:07-1:09) in That Day
The background flute sounds very similar to The Door motif part 1, it's only a small part that then trails off.
Now lets look at the notes. Disclaimer, I am not a musician in any way, but I feel like this is clear as day.
The first set of notes is The Door motif. The next two sets of notes are two different transcriptions of that part in That Day. While there is some error, these two transcriptions are also different from each other, I feel like this is natural when you are trying to reconstruct a song by ear. Despite the error, I feel like these are trying to communicate the same fundamental idea. They are still extremely similar. Even if they are different not because of error, the motif itself does not have to be 1:1.
Also think about the thematic similarity! Asgore is literally talking about the dark fountains in the scene where That Day plays! The Door is what plays when you see the chapters 1 and 4 dark fountains!
I feel like this is a very intentional reference and it should be added to the Leitmotif page already.