According to Wikipedia,
"In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set"
If we are using terms like "lying between two numbers", then why not define it as follows?
"Interval (bounded) is a set of numbers which contains all real numbers between two given numbers."
"An unbounded interval is a set of numbers which contains all real numbers greater than (or less than) a certain number,"
(And then specify closed and open intervals meaning)
Seemingly convoluted definitions usually exist because they're equivalent to the more intuitive versions and because experience dictates that they're easier to use in proofs (and/or less verbose).
For example, by not having cases the former definition you gave makes it easy to check when a subset happens to be a subinterval. You only have to examine that one property. Moreover, to some degree it is constructive in that it naturally lends itself to constructing the minimal (by inclusion) interval containing a set of points.
Your second definition has just enough complexity to make even those simple proofs a little annoying, and once you get the details worked out so that it's actually correct you'll be able to show the first is equivalent to the second anyway. Why not have the definition be the easy to use in the first place?