Your TLDR is slightly wrong. The Kalman filter is exactly Bayes theorem for linear dynamics and additive Gaussian noise. Bayes theorem is more general, but these assumptions yield a simple set of linear equations that can be solved very quickly.
In general, using Bayes theorem for filtering is often called the Bayes filter. Outside of a limited set of circumstances, it is not possible to implement. The Kalman filter is one set of circumstances because linear transforms of Gaussian distributions remain Gaussian, and we can specify a Gaussian process just by its mean and covariance. Another set of circumstances is when you have a discrete system with a small enough number of states.
In general however, the Bayes filter results in an infinite dimensional dynamical system since it yields a typically continuous function representing the distribution of your state given your measurements. The Kalman filter, particle filter, and other methods approximate this problem via a more tractable finite dimensional representation.
You are correct, but I don't get where I am wrong. Is it just that I didn't write "equivalent to the Bayes filter equations for linear dynamics and additive Gaussian noise", or do I miss something else?
In general, using Bayes theorem for filtering is often called the Bayes filter. Outside of a limited set of circumstances, it is not possible to implement. The Kalman filter is one set of circumstances because linear transforms of Gaussian distributions remain Gaussian, and we can specify a Gaussian process just by its mean and covariance. Another set of circumstances is when you have a discrete system with a small enough number of states.
In general however, the Bayes filter results in an infinite dimensional dynamical system since it yields a typically continuous function representing the distribution of your state given your measurements. The Kalman filter, particle filter, and other methods approximate this problem via a more tractable finite dimensional representation.