It seems that "bound vector" relates to a "specific force"
and "free vector" relates to a "general force".
If I talk about F(AB) = the force on "A" due to "B", then I am talking
about a push with a specific magnitude and a specific direction AND
located at a specific point on "A" (assuming a "point force" for
simplicity).
As far as solving the equations of motion, this is an over-specified
condition. Any OTHER force of the same magnitude and direction would
cause the same acceleration of "A". It doesn't need to be this specific
F(AB) (ie "bound vector"). It could any other similar force (ie free
vector).
So F(net) = ma works for ANY (free) vector, and it works in particular
for this specific (bound) vector F(AB).