In api , it is retreiving data on the behalf of term. Try send term instead of name or try this in index method of ProductController.
//name into term
$request = request()->all();
if(isset($request['name'])){
$request['term'] = $request['name'];
}
// category_id into category
if(isset($request['category_id'])){
$request['category'] = $request['category_id'];
}