you make valid points! however, you did misuse some terms (no shade though hehe)

"SI" is short for "système international d'unités", or international system of units. thus, we call it "the SI", not "SI system".

the SI outlines practices for using units, along with an approved set of units. they are the ones who say to write "3.2 kg" and not "3.2kg" or "3.2 Kg", for instance. the units outlined include stuff like the meter, kilogram, and second too—in fact the SI is the successor to the "meter-kilogram-second" system! however, the SI does not include units such as millibar or liter. hectare, for example, is a metric unit, but is not an SI unit. this wikipedia page has a big list of them.

basically, the SI is a standard used so scientists don't get confused by units.

the metric system, or rather a metric system, is a framework for attributes of stuff (aka numbers and units!) that is decimal-based (i.e. around powers of 10), uses a set of specific base units, and has ways of specifying unit multiplication applicable to any unit. the SI is a metric system!

while there are other metric systems too, most commonly you will see the spirit of the SI used in conjunction with non-SI units. this is what "the metric system" usually means. the liter is a very common example; the SI would say liter is not to be used and to instead use cm³, but you'll still see L and mL about. their connotations are useful!

but yeah honestly so long as they work and don't cause confusion use whatever units you want.