As clear from source,I wonder what's the need to use that in the mid name of Steven "Steve" Barry Smith.I have pretty much seen these on Wikipedia website.
Quotes in this way between a person's given and family names generally indicates a nickname (a name they are informally referred to)
John "JJ" Smith
Would be referred to as "JJ" in informal situations (as by close friends or family
The nickname also might be used for musicians, athletes, et alii.
James "Left-Hand" Robinson
might be the name of a baseball player known for his skill with his left hand and is thus referred to as "Left-Hand"
The quotes will signify a nickname. The person in your example has the legal name of Steven Barry Smith. "Steve" is a shortened, informal "nickname" for the name Steven.
In a formal setting, he would be addressed as Steven Barry Smith. In a more casual or friendly setting, people would call him Steve Smith.