I suspect that the police are partially correct with their interpretation but not with regard to signalling equipment
The definition of a "firearm" is contained in s 57 of the Firearms Act
57.
Interpretation.
— (1) In this Act, the expression “firearm” means a lethal barrelled weapon of any description from which any shot, bullet or other missile can be discharged and includes—
(a)any prohibited weapon, whether it is such a lethal weapon as aforesaid or not; and
(b)any component part of such a lethal or prohibited weapon; and
(c)any accessory to any such weapon designed or adapted to diminish the noise or flash caused by firing the weapon;
and so much of section 1 of this Act as excludes any description of firearm from the category of firearms to which that section applies shall be construed as also excluding component parts of, and accessories to, firearms of that description.
There was a case a few years ago in which it was held that a pistol flare fitted the definition of a firearm and therefore amounted to such under the relevant part of the Firearms Act. I think the case might have been called R v Singh
Note that section 13 of the firearms Act 1968 provides
13.
Equipment for ships and aircraft.
— (1) A person may, without holding a certificate,—
(a)
have in his possession a firearm or ammunition on board a ship, or a signalling apparatus or ammunition therefor on board an aircraft or at an aerodrome, as part of the equipment of the ship, aircraft or aerodrome;
(b)
remove a signalling apparatus or ammunition therefor, being part of the equipment of an aircraft, from one aircraft to another at an aerodrome, or from or to an aircraft at an aerodrome to or from a place appointed for the storage thereof in safe custody at that aerodrome, and keep any such apparatus or ammunition at such a place; and
(c)
if he has obtained from a constable a permit for the purpose in the prescribed form, remove a firearm from or to a ship, or a signalling apparatus from or to an aircraft or aerodrome, to or from such place and for such purpose as may be specified in the permit.
It follows that a certificate is not required for signalling equipment but would be required for bird scaring equipment assuming it fulfilled the definition contained in s 57.
Chevvron
The definition of firearm makes no reference to rifling therefore a smooth bore does not automatically make a gun a shotgun
section 1 (3) (a) of the Act defines a shotgun:
a shot gun within the meaning of this Act, that is to say a smooth-bore gun (not being an air gun) which—
(i) has a barrel not less than 24 inches in length and does not have any barrel with a bore exceeding 2 inches in diameter;
(ii) either has no magazine or has a non-detachable magazine incapable of holding more than two cartridges; and
(iii) is not a revolver gun