Yes.

2001 was when the Silverado HD and Sierra HD were formally introduced.

The only one that's iffy is the 2001-2003/2005-2007 1500HD (was dropped for '04 and brought back in '05). Sometimes it's in the brochure with the light-duty 1500 trucks, other times it's marketed with its bigger brothers.

Television commercials for the Silverado HD:
https://www.youtube.com/watch?v=JoZy1lY0n4E
https://www.youtube.com/watch?v=-yDOg7wUJ5k
https://www.youtube.com/watch?v=88DKWIlj01M

Brochures:
'03: https://www.motorologist.com/wp-content/uploads/2003-Chevrolet-Silverado-HD-brochure.pdf
'04: https://www.xr793.com/wp-content/uploads/2021/11/2004-Chevrolet-Silverado-HD-Spec-Sheet.pdf
'11: https://www.auto-brochures.com/makes/Chevrolet/Silverado/Chevrolet_US%20SilveradoHD_2011.pdf
'15: https://cdn.dealereprocess.org/cdn/brochures/chevrolet/2015-silverado2500hd.pdf
'22: https://www.chevrolet.com/content/dam/chevrolet/na/us/english/index/shopping-tools/download-catalog/08-pdf/2022-chevrolet-silverado-hd-ebrochure.pdf

I haven't looked into it but did Chevrolet market them as a separate model, like Ford and Ram do? And when did they start doing that?

I would say it's fairly important, as the HD trucks have an entirely different chassis and the codes to go along with it. For instance, we list Sierra Denali as a model name, which is not nearly as important as the distinction between a light-duty (1500) and heavy-duty (HD) model.

I don't know how important is that mention.
I'd say that it can go as extra info, but if it is an important distinction in the model it is not a problem to have it in the name.

I've noticed we have some (a minority) of Chevrolet Silverado HD models listed as the model name 'Silverado HD', while a majority of others have 'HD', '2500HD', '3500/3500HD', etc. in the extra info section. Should I put them all as model name Silverado with the 'HD' in extra info, or should I go through and change the model name on the others to 'Silverado HD' and 'Sierra HD' bring them more in line with how the Ford and Ram pickups are listed (F-150 vs. F-Series Super Duty/Ram 1500 vs. Ram Heavy Duty)?

I know there was a discussion about this a few years ago, but I don't remember what conclusion was reached, if any.