according to all the info I could find yes it was 2002 model year that the H1 name was adopted:

For instance this website http://www.h1owner.com/Changes/02changes.html when talking about changes to the 2002 H1 states:

"With the introduction of the H2 in 2002 (2003 model year), the original Hummer now carries H1 badges for the first time."