In a broader perspective, are SaaS providers, that inherently have customers all over the world, expected to register a trademark in all countries or only the countries they are supposed to have offices in?
Most of the time you can file for an international TM under the Madrid Protocol. But you can delay filing in most circumstances under the Paris Convention (generally for 6 months).
But you should consult a lawyer. It's very are for a startup SAAS company to file TMs in more than 3 or 4 countries.
Trademark doesn't provide a constraint on trade, and additionally most courts would find that "Notion" and "Notion Labs" are sufficiently different to not cause confusion (depending on the classes of TM each owns)
This is complicated and expensive, and depends on specific circumstances, as well as dates of filing.
Using them as a deterrent is a nice to have, in your key markets.
But if someone challenged that by using a similar name or casually turning your trademark into a verb, it becomes a war of attrition that you win by never playing.
(1) Where does country/jurisdiction X consider the Internet to be for the purposes of defining jurisdiction? Is it where the misled customers are? Is it where the infringer's servers are? What about Google? How will this affect your ability to enforce your TM rights, registered or otherwise?
(2) How does one acquire TM rights in country/jurisdiction X?
(3) What are the requirements of TM registration in country/jurisdiction X? Does it require "use"? What does "use" mean in country/jurisdiction X, including in light of (1) above? How long can you delay commencing "use" in X (or pause use in X) without losing your application/registration? How does X deal with Madrid Protocol filings? What priority, if any, will securing a registration give you over pre-existing but unregistered users of the same or a similar TM in X, and does this situation evolve over time? What is the practical cost of securing and maintaining a registration in country/jurisdiction X? Is your TM "confusing" with some other not-identical TM already used and/or registered in X? What does it mean for one TM to be "confusing" with another in X? Is your TM unregistrable in X? What does it mean to be unregistrable in X?
(4) What are the courts of country/jurisdiction X willing to do to enforce rights arising from a TM registration in X? If a court of X can't stop the infringer directly, would it consider enjoining a third party such as Google to at least stymy the infringer (e.g. by making the infringer unsearchable using Google)? What scope of protection are the courts of X willing to extend to your TM? Will the courts of X let other persons get away with minor variants? Will the courts of X let other persons get away with using the TM on unrelated goods or services? How unrelated? Will the courts of X recognize an order of a court of country/jurisdiction Y?
But no idea how well this works, and in practise I reckon you'd still want to register trough the Madrid system for the regions you plan to do business in directly
So far, the only tangible benefit of having the trademark was that we were able to claim a dormant Twitter account with our name. But I can imagine it can help in the longer run as well...
Names can be changed - the solutions you offer are more important.
And as SaaS provider it would be stupid from competitors to use the same name because you probably already own e.g. the domain => so it would be more hassle for them (except for the countries where fakes are common, but no trademark protects you in these regions)
Obtaining trademarks in every country is not worth efforts unless you are at really high growth.