That's what I meant by a low-level format, which I read overwrites everything with 0's.
This isn't really low level formatting. Low level formatting is an old thing for hard drives but is still done on floppies that use frequency modulation (FM) encoding. Here's how it goes.
The drive writes to the entire disk and rewrites it from scratch. This is the critical difference. It writes carrier patterns and track boundaries, not ones and zeroes.
Old drives had the facility to be aligned (with a screwdriver). Say, one drive had it's heads misaligned a little toward the centre of the disk and another a little to the outer of the disk (say, half of the track width). If you did a low level format on one drive and wrote some data, then put the floppy in the other drive it would read between the tracks and fail with that disk.
A floppy quick format instead skips the low level format and assumes it to be OK, then resets the file allocation table (a high level format).
With modern drives, the common alternative is the block check where data is written and read back to check that all is well. This is at the highest level.
The closest you can get to a low level format is resetting the partition table, and even this is not low level (well, lowish), and writing zeroes across the physical drive (beneath the partitions) is merely manually resetting the partition table.