Yes this is indeed what is referred to as a “leaky integrator”, as the discrete time approximation of such a device. The key element that makes it “Leaky” is having the positive real pole at $z < 1$ which is the equivalent of a continuous time real pole along the negative real axis.
With the pole at $z=1$, (and a zero at $z=0$) we get the discrete time approximation of a pure integrator, this is an accumulator with the transfer function given as:
$$H(z) = \frac{z}{z-1}$$
Any other constants added are gain scaling, but this represents an “integrator” equivalent as the impulse invariant mapping of $H(s) = 1/s$.
As we move the pole off of $z=1$ As given by:
$$H(z) = \frac{z}{z-\alpha}$$
With $\alpha$ as an arbitrary real constant between 0 and 1 (not using OP’s $\alpha$ but generally), it becomes a “leaky integrator” with $\alpha$ as the damping factor: the closer $\alpha$ is to 1, the less the “leak”. Any other constant as before is just a scaling factor.
As noted, this is the discrete time equivalent to moving the pole on the s-plane from $s=0$ that we get with $H(s) = 1/s$ into the left half plane to be:
$$H(s) = \frac{1}{s+ 1/\tau}$$
Where $\tau$ is the time constant indicating a decay given by $e^{-t/\tau}$ which comes directly from the inverse Laplace Transform of $H(s)$.