Where did j go in the underdamped response of an RLC circuit?
The general homogeneous solutions to a second order linear differential equation is the linear combinations of two linear independent solutions. The converse is also true: given two linear independent solutions, you can construct all other solutions from the two.

Therefore, you are free to choose the forms of these two solutions and these are just two of the forms:
$$ i(t)=e^{-\alpha t}(A_1e^{j\omega_d t}+A_2e^{-j\omega_d t}) $$
$$ i(t)=e^{-\alpha t}(B_1\cos(\omega_dt)+B_2\sin(\omega_dt)) $$
And you have demonstrated the first is equivalent to the second through linear combination by redefining the two constants.

For example, this is also a valid representation (but this would be a strange choice):
$$ i(t)=e^{-\alpha t}(C_1e^{j\omega_d t}+C_2\sin(\omega_dt)) $$