When ISPs promise you a certain speed that is to the internet, not to any point on the internet. The further you go, the more chances you have for congestion and speed reduction. So the Verizon speed test will normally show better results because they are testing to their server. Speedtest.net is going further on the net, increasing the opportunity for speed reduction.
Download speeds are dependent on the distance to the source server and that server's load. It is unlikely that those speeds will ever equal the promised speed. The 5.5mbs - 9.3mbs does seem low. But there are other factors beyond your connection that can influence it.
