Good summary, but you are confusing First Data's role - First Data is a processor, not just a Gateway, and is actually the processor for BAMS, Wells Fargo, and many other banks. (http://digitaltransactions.net/news/story/First-Data_s-Allia...)
Here are two articles for anyone who is interested in learning more:
[0] : https://www.quora.com/What-is-the-difference-between-a-payme...
[1] : http://www.chetu.com/blogs/finance-2/choosing-between-paymen...