Vancouver (via Hong Kong) to Shanghai (via Hong Kong) to Singapore

The most basic rule of 144 hour visa free transit states that only the immediate points before and after the mainland Chinese city matter. In other words, mainland Chinese authorities simply do not care if you make just a connection in Hong Kong or you will stay there for a week. In other words, your itinerary is, based on the basic rule:

Hong Kong - Shanghai - Hong Kong

As you can see, you do not have a country C. In other words, your itinerary is considered to be a return trip (round trip). Your country A is Hong Kong, your country B is mainland China and then you return to the country A. To conclude, this itinerary is not eligible for any kind of visa free transit.

However, you can make your itinerary eligible by introducing a third country, for example Macau. Like this you can use your first segment as it is until Shanghai, but you can buy a ticket from Shanghai to Macau for your return (second) segment. Like this, your itinerary will look like:

Hong Kong - Shanghai - Macau

When you land in Macau, you are free to return to Hong Kong immediately. Like this you have A - B - C sequence and your immediate points are located in two different countries or better to say in two eligible territories (eligible territories: Hong Kong, Taiwan and Macau) that act as third countries for 144 hour visa free transit purpose.

Thank you for the clarification and suggestion!

A follow up question: Must the air tickets from point A to Shanghai and from Shanghai to point C be issued as one connecting itinerary (i.e. under one record locator) or it is acceptable to have two separate one way tickets as long as the layover is less than 144 hours?

Separate tickets are perfectly eligible, so you can do that with absolute peace of mind. It is always the itinerary that is crucial in determining whether a passenger is eligible for 144 hour visa free transit or not. You are advised to print out your flight tickets ans well as your accommodation booking confirmation and to have them at hand.