I dont recall ever hearing or reading about steel core 22lr. generally steelcore is designed for armor piercing tendencies and a 22lr just doesn't have the force behind it to go through even a lowest level flak jacket.
however they may have made one much like the green tipped semi armor peircing ammo for pelt hunting so you wouldn't damage the hides of certain furbearers. I've never seen it but it's possible that it was made at one time.
in 1999 bill clinton signed a bill that made it illegal to sell surplus ammo, ammo intended for military use, and ammo designed to penetrate armor to civilians. semi armor peircing is still legal but all that black tipped ammo that flooded markets for years disappeared almost overnight. you can still buy it as long as it was manufactured before 1999. good luck for your search
EDIT, I finally got your link to work. basically they are outlining a bullet that is either a solid steel or copper core or it's weight is more that 25% steel or other hard metal for the jacket is considered AP. anything with less is considered a self defense or sporter round. so a a 40 grain 22lr bullet would have to contain less than 10grains of steel in order to not be considered armor piercing. it might be possible but 10gr is steel is not a whole lot, I dont know how much of a 62gr m855 is actually steel but I think a heavier 22lr like a 50gr would be possible but barrel twist rate would have to be more rapid than you standard 10/22 or marlin 60/795 I would think.
since 22lr is able to be used in a handgun a solid steel or copper core bullet would classify as a AP round
