How about this (non rigorous) definition appropriate for a multiverse.
An observer is ultimately just an algorithm that processes information. If you assume a multivere then the same algorithm will be implemented in amy different sectors. Time evolution is defined by the algorithm itself. I.e. if the algoritm is run then it maps itself to another algorithm.
Example: The precise way my brain works defines me. If you put all this information in a computer and write a program that simulates me then that program is ''me''. But if you run to simulate 1 second of time evolution then the algorithm effectively changes (because the brain isn't static). So, a new ''me'' is obtained that differes slightly from the old ''me''. The new version of me has subjectively evolved 1 second, even if the simulation of that single second took 1 billion years of computation.
In a multiverse setting all possible versions of me exist. Some are related to each other via ''subjective time evolution'' as described in the above example. So, the class of all versions of me can be split into subsets that can be ordered according to the subjective time evolution. Each version of me exists in the multiverse. But each element will have a memory of a ''previous'' version which is defined by this ordering.
Here I've ignored interactions with the environment, but that won't spoil this argument.
