Halifax is one that has a few different permutations.
At minimum would be this, which includes the historic main street and waterfront, financial area, legislature, and much of the oldest architecture.
In recent years, however, I'd make the argument that the commercial stretches along Water, Barrington and Hollis streets have become more or less continuous with the rest of downtown. The fact that they extend to the city's main train station (little-used though it is) and central market seem to bolster that. They also contain a similarly rich mix of 18th, 19th, and 20th-century architecture
The stretch heading west along Spring Garden Road to Robie has seen a dramatic spike in new development and tall buildings. Ten years ago it would have been more of an adjunct to downtown--an Annex type of neighbourhood. Today, the scale of development (and again, continuous mixed-use streetwalls connected to the rest of downtown) is quickly making it feel a lot more urban, extending west along this corridor and out of frame for a bit.
And finally, on the left side, Gottingen Street (where the arrow is) used to be a main commercial arterial branching north (this map is oriented with north on the left and east on the top). That neighbourhood was considered by many to be downtown, though in the 1960s the city destroyed a huge swath of the central city, severing the areas above and below Cogswell Street from one another. Cogswell is the red line marking the far left extent of downtown on the first three maps).
Gottingen sort of atrophied like a disused limb at that point, and I think it's fair to say that it can't really be considered downtown anymore, though some people will fight the point. It's seeing gentrification/revitalization now, however, so who knows in the future.