Both single ninja and team initiative matters. Initiative determines which characters get to attack first.

single ini means the ninja atacks first,team ini means wich passive field remains

So for example

i have ninjas 1 2 3 4 and enemy has a1 a2 3 a4

if my ninja 1 has 10k iniciative , and overall iniciative is 11k , while ninja a1 has 5k iniciative but overall iniciative is 15k

does it mean ninja 1 attacks first , but the ninja a1 barriers or buffs goes last? (that means the barrier for A team stays?)

Exactly.