Richard, Vincent, Katherine, Margaret, Lawrence, and Elizabeth have all been lost to me online because I have been looking to connect with Dick, Vinny, Kate, Peggy, Larry, and Beth - the names I know them as.
Just because your parents named you Stephen doesn't mean you are disrespecting them by being Steve online. And if everyone knows you as Steve, when they go looking for you online they're not likely to find you if you're listed as Stephen.
The Purpose is to Connect
The whole purpose of creating an online profile, on any social media channel, is to connect with people and to give them an authentic picture of who you are.
So the best approach is to think like a search engine. What will people search for when they look for you? Make that the name that you use on your social profiles. And while you're at it, make sure that the keywords that people are likely to think of when they think of you are also included in your profiles so they get picked up by search.
My real name is Jeffrey. It's the name my parents gave me and it is the name on my Passport and other formal/official documents. But everyone in my life knows me as Jeff. Fortunately for me, Jeffrey is likely to get picked up by most search engines when searching for "Jeff" but a scan of search results by any friend who is looking for me is likely going to miss my 7 letter full name if their brain is scanning for 4 letter names starting with J. So I use Jeff always.
Now imagine if your name is Margaret but you are known as Peggy! Who in the world is ever going to know to search for Margaret?
Respecting Your Formal Side
People aren't going to visit your profile repeatedly after they've connected with you. When they are looking for you you don't need them to find the formal you, just the you they are familiar with for one simple reason: so they can connect with you.
Include your more sophisticated name elsewhere in your profile if you can. In your Linkedin summary, maybe mention your full name somewhere or even tell the story of how you came to shorten your name if it's a good one.
You don't need to let go of your proper name. But if you don't use it regularly, put it in the same place you have it in your everyday life: in the background as a passing reference.
Photo ref: That's me and my friend Manami Fukuda being very sophisticated while modelling for a very fun and memorable event a few years ago.