To be precise, first-party and third-party just means whether the cookie set is for the domain you are currently on, or for another one. The latter do not have to be tracking cookies, but are often used as such. You can see the cookies that your browser is storing for a specific site by visiting it and looking at them in the developer tools (Storage or Application tab, depending on browser). Under the "domain" column you can see what domain it is for.

Furthermore, there you can look at the Local Storage and Session Storage tables which are also often used to store tracking data but are not prevented by cookie deletion.