A Site Template is essentially a WebTemplate, just pre-packaged. It's NOT a Site Definition.
If you would like to distribute it "globally" in your farm, crack the WSP open and edit the Feature and set the scope to Farm (instead of Site). (Quck and dirty solution)
Even better is to crate a Web Template of your own and copy and paste pieces from the Site Template that you exported - and imported into a new Visual Studio Project. I strongly discourage you to import the solution package and fiddle with that project and then re-deploy it. Been there, done that! That is - create a new project and only use the "imported" project as a temporary copy-and-paste solution.
Read more about Web Templates here: http://blogs.msdn.com/b/vesku/archive/2010/10/14/sharepoint-2010-and-web-templates.aspx