The correct definition of the power of a lens is as follows:
The power P of a lens is defined as the tangent of the angle δ by which it converges or diverges a beam of light falling at unit distance (1 m) from the optical centre (pole).
tan δ = h / f and if h = 1 meter, tan δ = 1/f or, δ = 1 / f
thus P = 1 / f
Thus the angle at which, a ray parallel to principal axis and 1 meter distance from it, is converged or diverged is the power of lens.
P = 1 / f is the way we calculate the power. It is not the definition.
The focal length is in meters. Unit of Power of lens is called Doptres. It is positive for the convex lens and negative for the concave lens.