You're right. It's just a white (or transparent white) drop shadow.
On that webpage, it seems to be a CSS drop shadow. That would be something like:
text-shadow: 0 1px 0 white; /* 0 x offset, 1px y offset, no blur, white in colour */
In Photoshop, this can be achieved many ways, but using a Drop Shadow Layer Style would be one of the better ways to do it.
In Illustrator and probably Inkscape, you could duplicate, change the colour and offset the text. It's worth noting that Illustrator does have a drop shadow feature: Effect > Stylize > Drop Shadow.
In GIMP? I have no idea. Duplicating, changing the colour and offsetting the text should work as well, assuming there's no similar feature to Photoshop's Layer Styles.