Congratulations! Very tidy code! 8)
If you can live with the fact that the Arduino is unresponsive for 15 sec after button1 or 5 sec after button2, you're done.
Thank you!
I certainly can live with that! It's serving a dumb purpose where it's going, anyways. It actually needs to be unresponsive, because it's essential no one makes the mistake of pressing the second button too early. If they do, they could screw up the measurements the device is taking, and render them useless. So it's not a bad thing, really! It's a safety 
Try pressing button 2 first.