Canadian citizen transiting/visiting multiple Schengen countries

When people talk about "transit", they are often referring to a flight itinerary that stops in an intermediate country for some period of time. For people who are from a country where a visitor visa is more difficult or more expensive, some countries offer an easier "transit visa" which allows people to stop in that intermediate country for a short time to make their flight connection.

None of the above applies to you. As a Canadian citizen, you are allowed to visit countries in the Schengen area for up to 90 days at one time. You are visiting Schengen for a week, no problem.

For visa purposes, it is often best to think of the Schengen Area as one, large country. Since you are Canadian, I would compare your European itinerary to something like:

• entering Canada in Vancouver by air (= arriving in Zurich)
• driving to Calgary (= driving to Berlin)
• flying to Regina (= your first outbound leg Berlin–Vienna)
• leaving Canada with a flight out of Regina (= your final departure from Schengen from Vienna)

The distances probably don't match with the European counterparts but the cities nonetheless serve as a nice exemplification.

In the Canadian example, your passport will be checked and stamped once in Vancouver (when you enter); likewise, in the Schengen Area this check will be in Zurich. I believe Canada does not check passports of leaving passengers (if it did, that would happen in Regina) but Schengen does which is why you will go through passport control again in Vienna.

Driving/flying across the province lines between Vancouver and Calgary or Calgary and Regina won't be of interest to anybody. Likewise, no government official will (usually*) want to see your passport on your drive to Berlin or the flight to Vienna, although you are crossing international borders.

For Schengen visa purposes, such itineraries are never 'transit'. For example, if you itinerary were actually non-Schengen–Berlin–Vienna–non-Schengen and this were a single air trip on one day, that would still be entry and exit. Likewise, if you were to fly Europe–Toronto–Vancouver–Japan, that is an itinerary that requires entering and exiting Canada. Some states have the concept of a transit visa, that allows a traveller to enter on one end of the country, gives them a certain amount of time to cross it and then exit on the other side. The classic case was Belarus which issued transit visas for people to travel from Poland to Russia by rail; you only had 48 hours between your entry on one end and your exit on the other. The Schengen Area does not issue such transit visas.

In air travel, transit often means arriving at an airport in a country that is neither your country of origin nor destination, spending a few hours at the airport and then taking a flight out. For example, you might fly Frankfurt to Seoul via Dubai; you would be transiting Dubai, never going through passport control and never leaving the airport. This does not exist in Canada as far as I know, as Canadian airports do not separate international from domestic gates (except US preclearance) so there is no 'transit zone' one might be in.

On the other hand, as Schengen requires exit immigration checks the flow of domestic (intra-Schengen) and international (non-Schengen) passengers can easily be kept separate; therefore, Schengen airports are attractive as a transit stop. Most nationals can travel on such itineraries (e.g. Istanbul–Frankfurt–New York) without any problems as long as the airline is willing to sell them. However, a few select nationals require an airport transit visa to do so. This does not apply to Canadian nationals. I suspect the primary purpose of airport transit visas is to prevent people from countries on that list buying a connecting flight through a Schengen country and then seeking asylum instead of continuing their journey.

One mistake many people make is to think that an itinerary non-Schengen–Berlin–Vienna–non-Schengen could be done in a set of transits, as if it were (say) Frankfurt–Dubai–Singapore–Sydney. However, this is where Schengen, the 'macro-country' kicks in: such an itinerary is functionally no different from non-Schengen–Berlin–Frankfurt–non-Schengen or Singapore–Osaka–Tokyo–San Francisco, and as the central flight is domestic you need to enter before and exit after it. You don't need to care about this, as you are describing a week-long trip which anybody would assume is entry and exit.

* Exceptions apply. For one, Germany may want to perform a quick document check which was introduced following the events of 2015. For two, current Covid regulations may or may not involve a brief check of documents. For three, you could technically be inspected by customs or checked completely randomly but I have never seen that happen in 20+ years of travelling across Schengen borders.