David, I don't think you'll have a problem. After your initial interaction with the web site, which is necessarily quite modern, your booking will have been put into other parts of United's reservation system, parts of which are probably very old and may be limited by things that haven't changed for years, such as perhaps the layout of the boarding pass as printed by different airport systems all over the world. AFAIR United's original reservation system called APOLLO started life in the 1970s; after they merged with Continental they decided to use Continental's system, SHARES, which began about the same time, though later I heard that they were trying to build bits of APOLLO into SHARES. The user-facing components will have been modernised but I wouldn't be at all surprised if parts of the back-office system still have limitations left over from when there were only 80 characters in a punched card!
Your full name will still be in there somewhere, but all concerned are used to names being compressed; as chrissw says, yours is quite straightforward compared to what happens to some people's names!