There are two options available when refreshing data. You can right click on the data source -> refresh, or you can right click on the data source -> extract -> refresh (there is an analog to this through the data menu as well). You want the latter. The former is more useful in a live connection context, but won't do anything for a WDC.
As long as you are choosing the extract refresh option, you are correct that Tableau will then launch the web data connector (in data gathering mode) to connect to your web data source and gather data.
Let me know if that helps,