My husband's family is from the Finger Lakes area, and we've spent several vacations traveling and hiking in the region.

iirc that architectural feature is called a widow's walk.

The 19th century homes in the area were very attractive to this Southern California native.

Those are very common in that area, and not just by the water. I was told years ago that the that the landowners would keep watch over their property from up there, as well. I've also heard it's a great spot to be if you're shooting squirrels. : )

The house is Italianate, but looks like it might have some earlier Greek Revival transitional elements. and the structure on the top is called a Belvedere.

A widow's walk would be a railed walkway around the belvedere, or a railed platform on top of the belvedere, or a railed platform on top of the main roof without a belvedere.

How do you get up to it? Does it allow light to flood down to the first floor through the stairwell? Or is it closed off, because I imagine it wreaks havoc with temperature regulation in the house.

In our friends' house with one, there are stairs from the attic into the belvedere/cupola. (We all call them cupolas here.)

What do you call it: Belvederes, Cupolas, or Lanterns?

They are often over a central staircase and let in lots of light. Yes, they can play havoc with temp control, but if you open the windows in the summer you get a nice flow of hot air up and out of the house.

Some have a glass pane between the turret and the main house, with the turret in the attic part. My house in VA had the remains of a cupola with a stained glass panel between cupola windows and the main staircase.

Alas, when the house was divided into apartments in the 1920s the stained glass was removed and the staircase closed up. It must have been lovely.

Sometimes I think they can create a light well into the house, but here there is usually some sort of skylight in the ceiling, it's not open to the house. I have been in houses where there are early light bulbs in the attic to light the skylight at night.

The one in my neighborhood has had all the windows removed and replaced with plywood and painted to match the trim. Apparently it leaked and they also had problems with window breakage with high winds and ice.

This article may help -
Life in the Finger Lakes - Classic

Around here (coastal) even the enclosed cupolas are called widow's watches or walks (people probably aren't up on their architectural lingo). There is an interior stairway up into them and people sometimes put a little bistro set or bench up there.