seems like these days most of the payment is 36% of ads in Google searches, but maybe there is also some money sent specifically for it to be the default.

They get paid a share of ad revenue their users generate if they make Google the default in all of their web search entry points. These are not separate. The deal is 1) "make Google the default" and then 2) "there's a revenue share". No 1 then no 2.