SPWeb.WebTemplateId..................

but is there a way to do this without writing
code? I see that in the template gallery, there is a system field
called site  definition ID ... It looks as though it applies to
templates rather than sites, though.

SPWeb mySite = SPControl.GetContextWeb(Context);
output.Write("<b>"+mySite.WebTemplate.ToString()+"</b> ");

mySite.WebTemplateId

rest of code

There is a column in the content database Web table. I think it is
called WebTemplate but I don't remember for sure.