Neither "micro" nor "macro", by themselves, have any real proper definition. They both are mearly vague terms implying the lens focuses closer than conventional. There have been atempts to define the term "macro" in the past, but it didn't stick.
The proper well defined terms are:
Photomicrography: Strickly speaking, photography through a microscope, but y extension, photography where the image on the sensor is greater than 10x the size in real life regardless of whether a microscope was used or whether the lens was mounted on a bellows or other extension device.
Photomacrography: photography where the image on the sensor is from 1x to 10x the subject's size in real life.
Macrophotography: the making of exceptionally large prints (e.g. wall sized murals) that are massively larger than the original image/negative/slide.
Microphotography: the making of exceptionally small prints (e.g. optically reducing an image of a circuit diagram onto an integrated circuit substrate).
With common camera lenses, "macro" is the common term for a lens that focuses very close. "Micro" is generally Nikon specific and has the same meaning. You often encounter the term "real" or "true" along with "macro" to imply that the lens focuses to 1:2 (half life size image on sensor) to 1:1 (life size image on sensor) rather than just to something in the 1:4 range (image on sensor is 1/4 life size).
A Japanese optical standards group, years ago, tried to force the 1:4 ratio as a minimum criteria for use of the term "macro". This stuck for a few years, but as more and more lenses came from non-Japanese sources, even those made by "Japanese" manufacturers, the rules faded out. Their justification for the 1:4 boundary was that a 4x6 print (the most common size) would be 1:1 compared to the original subject if a 35mm camera was used with a lens working at 1:4, a reasonable line to draw. An APS-c format (AKA "cropped sensor") camera would yield the same 4x6 printed image with only an optical magnification of about 1:6.