To describe the desired loop $\delta_\alpha$ in detail, let's look at the parts of the path $\gamma_\alpha \phi_\alpha \bar\gamma_\alpha$, and describe corresponding new parts which will be concatenated together to define $\delta_\alpha$.
The path $\gamma_\alpha$ goes along the bottom of the strip labelled $S_\alpha$ in that picture, starting from its initial point $x_0$. Define a correponding new path $\gamma'_\alpha$ which instead goes along the top of the strip, starting from its initial point $z_0$.
The closed path $\phi_\alpha$ goes around the boundary circle of the disc $e^2_\alpha$, which is where that disc is attached to $X$, and the base point of $\phi_\alpha$ coincides with the terminal endpoint of $\gamma_\alpha$. Define a corresponding new path $\phi'_\alpha$ which instead goes around a circle in the disc $e^2_\alpha$ that is concentric to the outer circle, and such that the base point of $\phi'_\alpha$ meets the terminal endpoint of $\gamma'_\alpha$.
Thus the concatenation $\delta_\alpha = \gamma'_\alpha \, \phi'_\alpha \, \bar\gamma'_\alpha$ is a closed loop based at $z_0$, and $\delta_\alpha$ is entirely contained in $A \cap B$.
Furthermore, letting $h$ be "the line segment connecting $z_0$ to $x_0$ in the intersection of the $S_\alpha$'s", the concatenation
$$\bar h \, \delta_\alpha \, h = \bar h \, \gamma'_\alpha \, \phi'_\alpha \, \bar\gamma'_\alpha \, h
$$
is a closed loop in $A$ based at $x_0$ that is path homotopic to $\gamma_\alpha \, \phi_\alpha \, \bar\gamma_\alpha$. That path homotopy may be seen geometrically as moving through the strip $S_\alpha$ and through the annulus joining the outer circle $\phi_\alpha$ of $e^2_\alpha$ to the concentric circle $\phi'_\alpha$.
It follows that $\delta_\alpha$ is a loop "in $A \cap B$ based at $z_0$ representing the element of $\pi_1(A,z_0)$ corresponding to $[\gamma_\alpha\, \phi_\alpha\, \bar\gamma_\alpha]$ under the base-point change homomorphism $\beta_h$".
I think it's not true in general that the orientation of attaching cells is irrelevant to the resulting attached space.
Consider the following example:
The following is 1-skeleton
I'm going to attach 2-cell as the following image.
There're essentially two ways to attach 2-cell. One is along $abab^{-1}cac^{-1}$ and the other is $aba^{-1}b^{-1}ca^{-1}c^{-1}$. Let $X_1$ be $X^2$ whose 2-cell is attached by the former way and $X_2$ be $X^2$ whose 2-cell is attached by the latter way.
Then, $\pi_1(X_1)=\langle a,b,c|abab^{-1}cac^{-1}\rangle$ and $\pi_1(X_2)= \langle a,b,c|aba^{-1}b^{-1}ca^{-1}c^{-1}\rangle$. Then $\pi_1(X_1)^{\text{ab}}=\Bbb Z_3\oplus\Bbb Z\oplus\Bbb Z$ and $\pi_1(X_2)^{\text{ab}} = \Bbb Z\oplus\Bbb Z$ so that $X_1$ and $X_2$ are not homeomorphic (even in the level of fundamental group).
Best Answer
The CW-flavoured argument uses the relative version of CW approximation:
Let $X$ be a CW complex, and $\iota \colon X^n \to X$ the inclusion of its $n$-skeleton (we omit the subscript on $\iota$ for the sake of notation). We want to show that $\iota_*\colon \pi_1(X^2) \to \pi_1(X)$ is an isomorphism. Suppose $S^1$ is given a CW structure so that the basepoint is a $0$-cell.
If $f\colon S^1 \to X$ is basepoint preserving, then by relative CW approximation there is a basepoint preserving homotopy between $f$ and a pointed cellular function $\tilde{f}\colon S^1 \to X$. By cellularity the image of $\tilde{f}$ is in $X^1$, so $\iota_*\colon \pi_1(X^2)\to \pi_1(X)$ is surjective.
Now suppose $f\colon S^1 \to X^2$ is a pointed map such that $[\iota\circ f] = 0 \in \pi_1(X)$, i.e. $[f]$ is in the kernel of $\iota_*$. Without loss of generality we can assume that $f$ is a cellular map. If we consider any basepoint-preserving null-homotopy $H\colon S^1 \times I \to X$ of $\iota\circ f$, then by relative CW approximation (note that $H$ is cellular on the subcomplex $(X\times \{0\}) \cup (\{x_0\} \times I) \subset X \times I$) $H$ is homotopic to a basepoint-preserving null-homotopy $\tilde{H}$ of $\iota\circ f$ which is cellular. In particular the image of this null-homotopy is in $X^2$ so in fact $[f] = 0 \in \pi_1(X^2)$, therefore $\iota_*$ is injective.
Note: an essentially identical argument shows that $\pi_n(X) \cong \pi_n(X^{n+1})$ for all $n\geq 0$, as an exercise you should write out the details.
Edit: Further note: after looking at Hatcher's proof of this proposition, it seems more elementary than the full force of CW approximation, even though I feel like CW approximation is the "conceptual" way to answer your specific question about CW complexes.