Describe the bug
I've been trying to hunt down an issue where a filter on wp-admin/edit-tags.php?taxonomy=post_tag&post_type=post URL expects data to come back as an instance of WP_Term but it instead comes back as stdClass, and I think it might be related to the format_hits_as_terms method in these lines. It looks to me like this method is returning $term as an instance of stdClass, never converting it to WP_Term even though it does use the standard formatting for a WP_Term object.

I first noticed this using The Events Calendar plugin because they have a call to the post_tag_row_actions where they force WP_Term on the parameter. But in any case, the WP docs specify WP_Term as the object type, too.

Steps to Reproduce

1. Index post_tag, at least, with some post tags.
2. Install The Events Calendar.
3. Go to /wp-admin/edit-tags.php?taxonomy=post_tag&post_type=post and search for a value that will return some tags.
4. The error I get is like this:

Fatal error: Uncaught Error: Tribe\Events\Taxonomy\Taxonomy_Provider::event_tag_actions(): Argument #2 ($tag) must be of type WP_Term, stdClass given, called in /wp-includes/class-wp-hook.php on line 307
in /wp-content/plugins/the-events-calendar/src/Tribe/Taxonomy/Taxonomy_Provider.php on line 53

Call stack:

Tribe\E\T\Taxonomy_Provider::event_tag_actions()
wp-includes/class-wp-hook.php:307
WP_Hook::apply_filters()
wp-includes/plugin.php:191
apply_filters()
wp-admin/includes/class-wp-terms-list-table.php:539
WP_Terms_List_Table::handle_row_actions()
wp-admin/includes/class-wp-list-table.php:1446
WP_List_Table::single_row_columns()
wp-admin/includes/class-wp-terms-list-table.php:345
WP_Terms_List_Table::single_row()
wp-admin/includes/class-wp-terms-list-table.php:253
WP_Terms_List_Table::display_rows_or_placeholder()
wp-admin/includes/class-wp-list-table.php:1289
WP_List_Table::display()
wp-admin/edit-tags.php:613

Expected behavior
I would expect to be able to search post tags, and I think in this case I would expect ElasticPress to return the term results as WP_Term objects.

Environment information

• Device: MacBook Pro (for my local environment)
• OS: macOS 12.4 (for my local environment)
• WordPress version:6.0.1 (for both local and remote)

• ElasticPress version: I'm running 3.6.5 that comes with WordPress VIP hosting in their mu-plugins folder, but I also see this bug on the develop branch.
• Elasticsearch version: 7.17.4 in my local environment. I'm not sure what version VIP runs.
• Where do you host your Elasticsearch server? I'm hosting mine locally, but I see the same issue on WordPress VIP.
• What ElasticPress features do you have currently active? Post Search, Protected Content, Related Posts, Terms, Custom Search Results

Additional context
I also tried locally editing the filter that The Events Calendar uses. I made it like this:

public function event_tag_actions( $actions, $tag ) {
    error_log( 'tag is ' . print_r( $tag, true ) );
    return $this->container->make( Event_Tag::class )->event_tag_actions( $actions, $tag );
}

Then when I run that log, here's what it returns:

[27-Jul-2022 21:49:54 UTC] tag is stdClass Object
(
    [ID] => 8134
    [site_id] => 1
    [term_id] => 8134
    [name] => 2008 election
    [slug] => 2008-election
    [term_taxonomy_id] => 214476
    [taxonomy] => post_tag
    [description] => 
    [parent] => 0
    [count] => 1
    [meta] => Array
        (
        )

    [elasticsearch] => 1
    [filter] => display
)