A true legal deep dive!
The FA 1968 has seen several through-life amendments. The most recent of which (the 1968 Act) was to subsume numerous other legislative provisions for Firearms/Shotguns/Guns/Rifles which were created in silos to make life easier for those enforcing the law. Previously, it used to be the case that you would have separate statutes for each type of 'weapon' (for want of a better term). This is very common for pre-20th century law, and you will see that modern-day legislation looks to try to cast a net over as many statutes in a bid to try and simplify the law both for the benefit of the lay-person and those who enforce it.
The FA 1937 (predecessor of the 1968 Act), encompassed something called the Gun Licence Act 1870. Within this Act, the term 'gun' is given a legal definition, that is:
'In this Act the term "gun" includes a firearm of any description and an air gun or any other kind of gun from which any shot, bullet, or other missile can be discharged..."
(It doesn't take a genius to work out why they didn't stick with this definition when they made the 1968 Act)
Clearly, as the times move on, so do legal definitions, which is why you will see in s57 FA 1968 (and in most post-1950 legislation) there is a section named 'Interpretation'. This is common nowadays as they aim to prevent confusion, often with limited effect, as proven here! Before people get annoyed at this, it's worth bearing in mind that this ambiguity could be advantageous to your case if you find yourself on the wrong side of the law.
From what I can see, "Gun" holds no real legal definition nowadays. I would make this assessment based upon the reason being that most of what we have can be accurately and legally categorised by use of s57 FA 1968.
Hatch