You're not confusing anything, it is the same disease.
However Shireen has only a relatively small scar not because it's an early stage, but because she is cured - see below.
Greyscale spreads through touch:
Greyscale is extremely infectious, spread through touch contact with
an infected person, or even spread on unsterilized objects that have
been touched by the infected.
People infected with greyscale eventually become Stone Men, unless they die earlier. It is also possible to recover, which is rare, but happens.
Those that manage to survive a bout with the illness will be immune
from ever contracting it again, but the flesh damaged by the ravages
of the disease will never heal, and they will be scarred for life.
This was the case with Shireen.
It takes a long time for an infected person to reach the "Stone Man stage":
Greyscale kills slowly, often taking many years to progress to a
terminal state. Long before that, in the mid-to-late stages of the
disease, the infection will spread to cover all of the victim's skin
from head to toe, leaving them in agony. An infected person can
survive like this for years, living in misery as one of the Stone Men.
In the very final stages of the disease, (perhaps years after the skin
is completely covered with it), it starts attacking the body's
internal organs, hardening them as it did the outer flesh. When it
spreads to the brain, the deterioration can result in violent
insanity.
