Final answer:
The Williams family most likely stayed in one city to see the sights while they were at Niagara Falls, where they spent several days exploring the area.
Step-by-step explanation:
The day on which the Williams family is most likely staying in one city to see the sights is when they found themselves at Niagara Falls. The excerpt provided indicates that they were "westward bound" and arrived at Niagara after traveling by slow conveyances of the period. They spent several days walking and driving around the falls.
Specifically, it mentions that one morning they stood on the Canadian side watching the waters. This indicates that they stayed in the vicinity of Niagara Falls long enough to explore multiple viewpoints. In contrast, the other mentioned snippets from their trip do not suggest a stay in a single location focused on sightseeing.