Domains: Use correct flowName for domain search component (#92097)

* Set `flowName` to `domains` once we navigate to `/domains/add/:site`

* Improve notice message

client/lib/domains/registration/availability-messages.js

@@ -560,7 +560,7 @@ function getAvailabilityNotice(
 
 		case 'gravatar_tld_restriction':
 			message = translate(
-				'Gravatar is currently offering free .link domains. Additional domain extensions may become available for a fee in the future.'
+				'Gravatar is currently offering .link domains. Additional domain extensions may become available for a fee in the future.'
 			);
 			severity = 'info';
 			break;

client/my-sites/domains/controller.jsx

@@ -74,6 +74,7 @@ const domainSearch = ( context, next ) => {
 				<DomainSearch
 					basePath={ sectionify( context.path ) }
 					context={ context }
+					isAddNewDomainContext={ context.path.includes( 'domains/add' ) }
 					domainAndPlanUpsellFlow={
 						context.query.domainAndPlanPackage !== undefined
 							? context.query.domainAndPlanPackage === 'true'

client/my-sites/domains/domain-search/index.jsx

@@ -46,6 +46,7 @@ import getCurrentQueryArguments from 'calypso/state/selectors/get-current-query-
 import getCurrentRoute from 'calypso/state/selectors/get-current-route';
 import isSiteOnMonthlyPlan from 'calypso/state/selectors/is-site-on-monthly-plan';
 import isSiteUpgradeable from 'calypso/state/selectors/is-site-upgradeable';
+import { setCurrentFlowName } from 'calypso/state/signup/flow/actions';
 import { getDomainsBySiteId } from 'calypso/state/sites/domains/selectors';
 import {
 	isSiteOnECommerceTrial,
@@ -121,6 +122,10 @@ class DomainSearch extends Component {
 		}
 		this.checkSiteIsUpgradeable();
 
+		if ( this.props.isAddNewDomainContext ) {
+			this.props.setCurrentFlowName( 'domains' );
+		}
+
 		this.isMounted = true;
 	}
 
@@ -449,5 +454,6 @@ export default connect(
 	{
 		recordAddDomainButtonClick,
 		recordRemoveDomainButtonClick,
+		setCurrentFlowName,
 	}
 )( withCartKey( withShoppingCart( localize( DomainSearch ) ) ) );