SearchBoxScriptWebPart and custom display template

We are rendering search box dynamically in a child site using the below embed:

Microsoft.Office.Server.Search.WebControls.SearchBoxScriptWebPart, Microsoft.Office.Server.Search, 
Version=15.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c|ResultsPageAddress;https://*****.com/search/Pages/results.aspx,
TryInplaceQuery;false

We have implemented a custom display template that is being used by search control in parent site. If I don't provide the rendertemplateid property, the dynamic control reverts to custom display template. Shouldn't it revert to native display template?

Also, what is the difference between Control_SearchBox_Compact or Control_SearchBox display templates?

Thanks!