Given that Google, the most well known and used search experience trialed and killed instant search (Google instant) and relies on a 2 step search requiring users to type their query and then search in combination with auto suggest and previous searches - is this the most expected search pattern?
Users love search for two reasons(Nielsen):
- Search lets users control their own destiny and assert independence from websites' attempt to direct how they use the web.
- Search is also users' escape hatch when they are stuck in navigation.
So if you want to tame a lot of dynamism you always need to go for Autosuggest and previous search queries which can also limit the endless frustrations we have from search. As it can be the source of the worst usability problem on the web.
So these can be thought of for search experiences
Is the search providing more of answers than choices
Is it implicit within the product experience
Adapting to all form factors
Its more of a enabler to your users
To answer your own question you should search for studies and articles regarding user expectations for search that were conducted after the advent and public release of Google Instant and that take Google Instant into account.
Here is one I found using Google Search: https://npr.design/im-feeling-lucky-building-great-search-experiences-for-today-s-users-7129ae15cfa
I won't attempt to speak for the masses, however I can make an educated guess that: the most common behaviour of search mechanisms is also the behaviour that most people expect.