Based on Mrs. Blackwell's text to accompany this plate, I'm not 100% sure that this is the Summer Damask, though she identifies it simply as R. damascena.

"1. This Rose Bush grows not so tall as the white, but taller than the Red; the Leaves are a light Grass Green, and the Flowers a pale Red.
2. It grows in Gardens and flowers for several Months in the Summer.
3. The Flowers are of a gentle cathartic Nature, purging choleric and serous Humours. They are frequently given to Children & weakly Persons, mixt with stronger Cathartics.
4. Greek rhodon. Latin, Rosa damascena vel pallida Spanish, Rosas, Italian Rosa. French, Roses. German, Rosen. Dutch, Provincie Roos."

Though she was writing for apothecaries, and was not an expert in roses per se, I find it curious that she said that it "flowers for several Months in the Summer," since no once-blooming Damask is known to do that.