Giving this some thought*, I realised that the test environment having to run on the Vagrant box, while the test itself was running on the Mac was the likely culprit. The tutorial stresses that it is important that the tests are ran with a user with the proper permissions. Presumably, because it will need to initialize the environment which will then need to be accessed by the webserver to run the actual test. So, what do you do? Install phantomJS on Vagrant? Sure, that would have worked, most likely. But what worked too, was just setup a little port forward from port 8510 on the Vagrant box to the same port on the host:
ssh -L 8510:192.168.33.1:8510 localhost
This is issued on the vagrant box. You may need to change the IP address of your host, if it is different, and you will need to provide the password for the vagrant account on the vagrant box (which usually is just "vagrant").
I both love it and hate it when the solution to a problem is a one liner.
*: This sort of insight usually doesn't come to me at the end of a long day of hard work, but instead in bed, in the shower, or, in this case, on the Lufthansa flight to Amsterdam while gazing out of the window at all the pretty lights below.