The May 23 update is actually not for Note 3, but also for Galaxy Note 2, Galaxy S4 and Galaxy S4 Mini devices. The Android update will come to the Galaxy S4 smartphones five days after, which will be on May 28.
Samsung has started rolling out Android 4.4 KitKat update for GN 3 in January, followed by an update for Galaxy S4 after a month. The updates began to spread in devices worldwide.
This May has seen more Android updates for Note 3, specifically in Orange, France, Meteor, Ireland, Portugal and Brazil. They either got the update via over-the-air (OTA) or Samsung Kies. Some carriers have announced the updates to their subscribers while others are holding it for specific release dates. Verizon Wireless is one of those.
While known to be almost always the last carrier to push for updates to its Android users, Verizon does an extensive testing before releasing updates to guarantee stability. The Big Red carrier does not want to send broken builds to its users.
This strategy at times goes against Verizon as many customers transferred already to other carriers because they do not receive major updates soon. But there are many cases that other carriers have to stop various updates rollout because of bugs. Verizon wants to play the "better safe than sorry" game with its subscribers.
Galaxy Note 2/3 and Galaxy S4/S4 Mini owners should start receiving their Android 4.4.2 KitKat upgrades from the Big Red on the cited dates through OTA deployment.
There is no guarantee how the carrier will finish its thorough testing of the Android 4.4.2 KitKat update for Galaxy Note 3 and the other three Samsung devices.
Thus the cited dates should be taken lightly until Verizon itself will announce the Android updates. It usually does so in its official changelog for its smartphone portfolio prior to the actual release.