There is really no difference. You can frame the Kalman filter as a Bayesian posterior inference problem.
For example, for a stationary linear Gaussian model, you have a transition model of the form:
z_t = Az_{t-1} + Bu_t + e where e ~ Gaussian(0,Q)
and an observation model of the form:
x_t = Cz_{t} + Du_t + d, where, d ~ Gaussian (0,R)
Since, z_t and x_t are both multivariate gaussians in this model, you can compute the posterior distribution on z_t's, which will also be a Gaussian. That is basically the Kalman filter.
As the writeup mentions, you might choose a non-Gaussian noise model, in which case the posterior distribution is not a Gaussian and then you employ something like a unscented Kalman filter or extended Kalman filter.
In the event of Gsussian noise, linear state transition and linear measurement, KF yields the exact posterior.
Both unscented and extended refer to non-linear measurement or state transition equations. Neither are designed for non gaussian noise.
Though the KF and it's variants are one of the simplest, well-performing estimation methods out there, so it wouldn't suprise me if it's used for everything, appropriate or not.
For example, for a stationary linear Gaussian model, you have a transition model of the form: z_t = Az_{t-1} + Bu_t + e where e ~ Gaussian(0,Q) and an observation model of the form: x_t = Cz_{t} + Du_t + d, where, d ~ Gaussian (0,R)
Since, z_t and x_t are both multivariate gaussians in this model, you can compute the posterior distribution on z_t's, which will also be a Gaussian. That is basically the Kalman filter.
As the writeup mentions, you might choose a non-Gaussian noise model, in which case the posterior distribution is not a Gaussian and then you employ something like a unscented Kalman filter or extended Kalman filter.