Good story. Keep up the good work

Great Story. Hope to see some more of this

Really great work

Uh, why are those two specific Limit Breaks Noble Phantasms? I mean they are essentially just a luck boost and an, admittedly powerful, sword skill. I get why Chain Slash, Summoning and Chocobo Mode aren't Noble Phantasms. But Rush Assault and Meteorshots are both incredibly powerful unusal attacks, the former even making his eyes and fists glow. Healing Wave grants him incredibly powerful healing abilities, Air Strike litterally summons Helicopters out of freaking nowhere too shoots his enemy and Apocalypse... Well suffice to say, if only one of Zack' Limit Breaks became a Noble Phantasm, I would expect it to be Apocalypse. Also, if you don't mind me asking, how exactly is the DMW gonna work if some Limit Breaks are downright Noble Phantams and others are not?
Honestly, if you are gonna pick just two LBs to be NPs, I would have chosen Apocalypse and Healing Wave with either Rush Assault or Meteorshots as a close third. Personally though (and this is just if it were me, you don't need to go along with it) I would have made both the Buster Sword and the DMW Noble Phantasms (D-Rank for the Buster Sword and either E-A()() or C-A()() for the DMW) and try to come up with more fitting abilities for his skills. Maybe even just pick some of the more common personal skills canon to Fate that would fit Zack's skill set like Eye of the Mind or Battle Continuation or something along those lines.
PS: All of the western Servant's NPs and even those eastern which don't use Kanji or similar lettering in it's name (basically all but most of the chinese or japenese Servants NPs) have a title or second name (ex: Exalibur: Sword of Promised Victory). Are you gonna come up with a title for Zack's NPs?

Zack has DMW ranked A. Does that mean that he can use summons?
If he can use them. Try to use a mixture of popular summonses like Bahamut and lesser use ones like Tonberry.
Other than that. This is a good opening.