The reason you find wildly varied depictions of Faust is because there are quite a few versions of the Faustian legend, Goethe's being fairly recent. Here's a brief and incomplete list:
Artistic depictions of the character may be based on any of the countless versions of the legend. And of course as with any fictional character some artistic license can be assumed, you can't really expect complete historical accuracy in art.
If you are interested in Goethe's version and assuming Goethe followed contemporary fashion in his works, you'll need to research German fashion in the late 18th and early 19th centuries. This image, from the Wikipedia article on 1795–1820 in fashion, shows a German gentleman in 1815:
Goethe's inspirations for the play included Jacob Bidermann's (1578 – 1627) Legend of the Doctor of Paris, and that may also be a timeframe you'd be interested in looking at. However, since Goethe doesn't mention the exact timeframe Faust takes place in, we can only speculate.
The "medieval wizard" image is quite common, and it's probably in reference to Johann Georg Faust (c. 1480 – c. 1540), an alchemist, astrologer and magician, the suggested real person behind the legend. Your second picture is actually a 17th century portrait of him.