As background, there are functions in WordPress that tell you certain things about the main query. For example, is_search(), is_archive() etc. will tell you whether the main query is a search query, or an archive query respectively. These typically, but not necessarily, line up with where certain templates will be used.
You use these functions if you want to know this information about the main query, and therefore know what kind of page you’re looking at. However, note that the main query is an instance of WP_Query, but developers can create their own queries by creating their own instances of WP_Query.
Instances of the WP_Query object have equivalents of these conditional functions as methods. You can use these methods if you want to know the same thing about any of those specific queries, even if it’s not the main query. For example (and assuming we are not on the search page):
$query = new WP_Query( array( 's' => 'Search term' ) );
var_dump( $query->is_search() ); // true
var_dump( is_search() ); // false
In fact, most conditional functions related to queries are just aliases for calling those methods on the main query. For example, is_archive() is exactly the same as this:
global $wp_query;
return $wp_query->is_archive();
So you use those functions as methods (i.e. with $query->) if you need to know if that query matches that condition. Or you use the naked function to check if the main query matches that condition.
Note that there’s nothing special about $query->. $query is just a variable containing an instance of WP_Query. So when doing that check you need to substitute $query with an instance of WP_Query
When it comes to using pre_get_posts, be aware that this action is applied to all queries on a page, not just the main query. When you add an action to that hook, the query that the hook is currently being applied to will be passed to the callback as the first argument, so you should check the conditions against that variable to only apply changes to queries that match the condition.