Yes it does.
The rule is:
if a player signs their ELC at 18, their contract can slide for 2 years. (Honzek signed at 18)
if a player signs their ELC at 19, their contract can slide for 1 year, unless they turn 20 that calendar year, then it doesn't slide at all because the signing age is considered to be 20.
Honzek in the AHL will slide if he doesn't play 10 NHL games.
Brzustewicz in the AHL won't see his slide at all because he signed at 19 and turns 20 before the end of the year.
edit: didn't see your edit, lol