Albumin is a type of protein found in large amounts in the blood. Because it is a small molecule in size, it is one of the first proteins able to pass through the kidneys into the urine when there are kidney problems.
This presence of small amounts of albumin in the urine is the condition called microalbuminuria. As kidney damage progresses and the amount of albumin in the urine increases, the name of the condition changes from microalbuminuria to albuminuria (macroalbuminuria) or proteinuria.
The key to remember is that there are other proteins besides albumin involved, especially later on. Early of its mostly albumin because of its small size. Later on its lots of albumin but also other proteins.
Microalbuminuria is now called moderately increased albuminuria while albuminuria (macroalbuminuria) or proteinuria is now called severely increased albuminuria.
Reference
https://my.clevelandclinic.org/health/articles/urine-protein-test