That is the old/other "9 game" rule. Under the new CBA, a player's contract can slide up to 2 years, regardless of whether they play in the AHL or not, so long as they don't play 10 or more NHL games.
If a player signs their ELC in X year, and if on Sept 15. of X year, that player is 18 years old, then he qualifies for the 2 year slide. So, even though Mantha signed his ELC in October of 2013 when he was 19 years old, he still signed the deal in 2013. And on Sept. 15 of 2013, Mantha was 18. Thus, for the purposes of that rule, Mantha signed his ELC as an 18 year old and thus qualifies for 2 consecutive years of his contract sliding, regardless of how old he is now and regardless of whether or not he plays in the AHL.