This question touches on multiple topics.
Log handlers (of any flavour) provide a standardised interface for you to log messages irrespective of the actual data sink (aka target). For more background read here
The two LogHandlers you reference just use different sinks. One logs to USB Serial while the other logs to SD.
So if you want to use any of the
Log.xxx() functions to dump your logs on an SD card you need to use the
SdCardLogHandlerRK library (which is internally using the
SdFat library to communicate with its desired data sink).
Log handlers also add some extra info to your data (e.g. timestamps, log scope, severity, …) which may not be desirable in your use case.
If you want to just read/write raw data (especially binary) from/to an SD card neither of the LogHandlers would help you with that. For that you should use the
SdFat library directly (without any wrapper).