LC Studio SD card adapter. This one uses resistor voltage dividers, which is not recommended, I know. But I think in this case it is irrelevant, because it was a problem not related to unclean signal edges
The LC studio card does not use resistor voltage dividers. The resistors are pull-ups. See This.
The full 5V Arduino output is applied to the signal pins of the SD. Most SD cards have input protection and sink enough current to pull the voltage down. Many cards sort of work with the LC Studio module.
Your "fix" for the LC Studio module won't work reliably with all SD cards. Bests to trash the LC studio module.
The problem is that even though the SD CS pin is high, the data on the SPI MOSI bus, which is intended for the display, still somehow confuses the SD card, even before we run SD.begin
This happens because of the poor design of the LC Studio module.