differential geometry - Inconsistency between $d_A = d + A wedge$ and $d_A = d(..) + [A,..]$?

I am confused by something basic stated in this https://physics.stackexchange.com/a/429947/42982 by @ACuriousMind and some fact I knew of. Here $d_A$ is covariant derivative.

1. 
   

   $d_A A=F$ --- @ACuriousMind says "The field strength is the covariant derivative of the gauge field."

   
   


2. 
   

   The Bianchi identity is $d_A F=0.$

   
   

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• In the 1st case, we need to define

$$d_A = d + A \wedge \tag{1}$$

So

$$d_A A= (d + A \wedge) A= d A + A \wedge A$$

• In the 2nd case, we need to define

$$d_A = d(..) + [A,..] \tag{2}$$

So we get a correct Bianchi identity which easily can be checked to be true

$$d_A F= d F+ [A,F]= d (dA+AA)+[A,dA+AA]=0$$

However, eq (1) and (2) look different.

e.g. if we use eq(2) for "The field strength is the covariant derivative of the gauge field.", we get a wrong result

$$d_A A = dA + [A,A] = dA \neq F !!!!$$

e.g. if we use eq(1) for "Bianchi identity", we get the wrong result we get

$$d_A F= d F+ A \wedge F\neq 0$$

my puzzle: How to resolve def (1) and (2)?

Could it be that for the $p$-form

$$d_A \omega = d \omega + \dots,$$ where $\dots$ depends on the $p$ of the $p$-form? How precisely?