I believe the book is wrong.
You should gain additional weapon training for free every tier (called Skill with Attacks). On the book, that ability is listed as a Maneuver for every tier other than third, but should actually be an ability gained every tier.
Why i believe it is wrong? Because Skill with Attacks as a maneuver is terrible on every tier compared to every other maneuver option available for your character.
At 6th tier you have a maneuver that allows your character to act a second time (or third/fourth if you can pay the cost), or you can pick an additional weapon training that will grant you +3 on your checks with a specific type of weapon, which is no better than the one gained at 1st tier. Nope.
And this is also the case for Numenera and glaives (the setting's equivalent to vectors), they gain Skill with Attacks every tier for free. The classes between the two books are not exactly the same, they have different abilities and flavor, but they are mechanically equivalent.
For weavers, it does make sense being a maneuver, and they can only pick it twice (3rd and 6th tiers), which also happens to Numenera's jacks on the very same tiers.