The word for a different press run (or print run) of the same edition is "impression", usually called "printing" in the United States. So you can have the first or second impression of the first edition, for example.

But be careful here. If this refers to an older book printed with movable type, you may be dealing with different editions. Different impressions are printed with "substantially" the same type. But before modern printing techniques introduced in the late nineteenth century, printers set up the type in a frame called a forme, printed the number of copies they thought they needed, and then dismantled the form to reuse the type for something else. If more copies were needed, they had to set up the type all over again. So different "impressions" (or "printings") of the same edition are really a modern phenomenon. If there are differences in the content, even small ones, we may be talking about different editions. In earlier books (before the late nineteenth century) it was quite common to have nine or ten different editions in a year.

All this is explained more fully in the following:
http://www.peterharrington.co.uk/blog/what-is-a-first-editio...

"The classic explanation of edition was given by Fredson Bowers in Principles of Bibliographical Description (1949). Bowers wrote that an edition is “the whole number of copies printed at any time or times from substantially the same setting of type-pages,” including “all issues and variant states existing within its basic type-setting, as well as all impressions.”"
https://en.wikipedia.org/wiki/Edition_(book)#Print_run