Confusing matrix elements

When we discuss about the matrix element of optical transition, we often can use either of the following ways:

1) H₁=exE, where e is the electronic charge, E is the electric field
2) H1=epA/m, where p is the momentum of the electron and A is the vector potential.

Following approach 1):

transition rate:
W=|<φ_i|exE₀|φ_f>|² δ(E_i-E_f-ħω)=e²E₀²|<φ_i|x|φ_f>|²δ(E_i-E_f-ħω)

Power:
P=ħωe²E₀²|<φ_i|x|φ_f>|²δ(E_i-E_f-ħω)

Dielectric constant (imaginary part):
ε₂=P/(ε₀ωE₀²)=ħe²/ε₀|<φ_i|x|φ_f>|²δ(E_i-E_f-ħω)

However, if we use the approach 2)

transition rate:
W=|<φ_i|epA/m|φ_f>|² δ(E_i-E_f-ħω)=(e/m)²A₀²|<φ_i|p|φ_f>|²δ(E_i-E_f-ħω)

Power:
P=ħω(e/m)²A₀²|<φ_i|p|φ_f>|²δ(E_i-E_f-ħω)

Dielectric constant (imaginary part):
using the relation that E₀=iωA₀
ε₂=P/(ε₀ωE₀²)=ħ(e/m)²/(ε₀ω²)|<φ_i|p|φ_f>|²δ(E_i-E_f-ħω)

We can easily see that the function ε₂(ω) in the two cases are very different, which one is correct and why?

In all the books I read, approach 2) seems to be used, why?