Yes, your problem is definately fuel related. However; before you spend $700+, first change only the "Fuel Filter" and see if that solves the problem. Always start with the cheapest solution first and then progress to the more expensive later if problem still exists. Remember, a professional mechanic is also a professional businessman in the business to make money (errr.....take your money). Thus; only change the "Fuel Filter" at this time, and then re-evaluate symptoms!!!
In addition; before starting your VX each time, it's always a good habit to first turn your key to "ACC" and then pause-n-wait until the "Fasten Seat Belt" idiot light goes out-- then fully turn the key and start the VX. I do this method everytime before start-up and it fires up everytime without fail. The fuel pump is "electric", and this start-up method helps prime the fuel system before iginition. Give it a try.
oh, and I already had them get the new pump. Should I ask to see the old one when they are done the work?