Is heart block the same as rate related left bundle branch block?

Not the same. Both are types of fatigue in the heart's wiring system. 1st degree heart block is when there is a delay in the signal getting from atria down to ventricles. That's minor. In more advanced heart block, some beats dont get through at all. In lbbb, there is no block between atria + ventricles but there is a delay in the signal spreading within the ventricles, which doesnt cause symptoms by itself.
Heart block and LBBB. Left bundle branch block is a form of arrhythmia or "heart block" as you mentioned. "heart block" would refer to all rhythm abnormalities, whereas left bundle branch block is a more specific type of arrhythmia.