Fix bug when a token is given public only (#32204) (#32218)

Backport #32204

routers/api/v1/org/org.go

@@ -191,7 +191,7 @@ func GetAll(ctx *context.APIContext) {
 	//     "$ref": "#/responses/OrganizationList"
 
 	vMode := []api.VisibleType{api.VisibleTypePublic}
-	if ctx.IsSigned {
+	if ctx.IsSigned && !ctx.PublicOnly {
 		vMode = append(vMode, api.VisibleTypeLimited)
 		if ctx.Doer.IsAdmin {
 			vMode = append(vMode, api.VisibleTypePrivate)