The issue of Cauchy's understanding of continuity is a subject of lively historical debate. Grabiner represents only one view in this debate. Laugwitz has published a series of scholarly articles studying the issue, including this 1987 publication in Historia Mathematica.
No discussion of this issue is complete without mentioning Cauchy's article dating from 1853 where he deals with convergence of series of functions and acknowledges that the condition as stated in 1821 needs to be modified.
Here Cauchy seems to introduce a condition close to uniform convergence which is stronger than one used in his earlier works. See for instance this 2013 publication in Foundations of Science and this 2018 publication in Mat.Stud..