I think that's right. 1st Foot Guards received the title 'Grenadier' and adopted the bearskin exclusively after Waterloo. Prior to that time only the grenadier companies wore the bearskin, which was customary throughout the infantry. The other companies wore the hat and later the shako.

Infantry drummers also wore the bearskin and fusiliers, already mentioned, had a slightly smaller version. I think it is generally accepted that bearskins, probably because of their expense, were not usually worn on campaign when they were replaced by hats or shakos.

The introduction of bearskin caps for grenadiers (and drummers and fusiliers) seems to have been with the 1768 Clothing Warrant because the preceding 1751 Clothing Warrant describes grenadier caps with the front part in the facing colour of the regiment and the rear red, that I take to mean the 'mitre' type.

The 1802 changes to the 1768 Clothing Warrant describe the same bearskin caps but notes that felt and leather caps (meaning shakos) replace hats, including for Privates of the Guards, and that grenadiers are allowed to wear them occasionally.

Here are some useful links.

http://fifedrum.org/crfd/1751.htm
http://www.fifedrum.org/crfd/1768.htm
http://royalscots.tripod.com/sitebuildercontent/sitebuilderfiles/1802clothingregs.pdf

JC