I suppose not, since both of those examples would be considered long-term memories. Though there are other ways to categorize memories though, like episodic memory, or semantic memory. I don't think we can conclusively say why some memories are affected in disease and others aren't, but there is an idea that some neurodegenerative diseases (like Alzheimer's disease) don't progress evenly throughout the brain, and might start in a specific area and progress into other areas later. So if there are some areas of the brain that are more important for certain types of memories, and the disease develops there first, then those will be the first memories affected. For example, if a disease starts in the hippocampus, which is particularly important for place memory, then maybe the first noticeable memory problems would be increasingly forgetting where things are.

Memory isn't my field of expertise, but one idea that should help you think about this question is that "electrical activity" in the brain isn't like electricity in computers, it isn't free electrons flowing along metal wires. It's electrochemical. Potential differences are built up by separated concentrations of ions in solution (e.g. Na+, Ca2+, Cl-). If you freeze the brain, presumably all those ions stay where they are in the frozen solution, so voltages persist and would still be there when thawed. It's like if you were to freeze a battery -- after you thawed it, wouldn't it still have a charge? So, ignoring any damage that freezing/thawing would have on the brain, based on our current understanding, I believe the mind would persist.

You might then wonder, even if it's not exactly electrical, what if all "electrochemical activity" were turned off? Perhaps by having all the ion concentrations equilibrate until there were no more potential differences? And to that, I don't think there is a solid answer, the neural mechanisms of memory are still hotly debated. People have observed memory engrams (i.e. a biological change associated with the formation of a new memory) in the form of "temporary" brain activity, while others have observed more stable changes in things like new synapses. An old idea is that short-term memory could stored by "activity" and long-term memory stored by these more stable changes, and if that were the case, then shutting off all brain activity would eliminate short-term memory but wouldn't affect long-term. But that's not a consensus view, so I don't think we can answer this yet.