How do coil overs adjust?

[ccohen5]

I am about to alter my coil over adjustment nuts for the first time and was
searching the web for the best tool, and any BTDTs.  One Miata site says
that you wrench up the shaft to raise the height, and wrench down to lower.

This is counter intuitive: if you raise the nut, you compress the spring,
and the car should sit lower. Vice versa if you lower the nut.

Where am I going wrong here?

I have only seen an after market tool at one after market site.  It has a
variable diameter that cannot be locked. So each time you engage the pawls
of the adjustment nut, you have to reset the size of the wrench claw.

Colin