I have been seeing the following on Facebook under posts:
Like · Comment · Share · Website
Like · Comment · @GThomasJewelers on Twitter · Share
How do you do this—have a website or Twitter link under every post?
Those posts are marked as such because they were originally tweeted. If you link your Twitter account to Facebook, your tweets will appear under Facebook as shown by you.
You can connect your Twitter account to Facebook from your Twitter profile settings.