Hyperbolic lens shape and aspheric surfaces In Optics, Hecht, the author states that the perfect surface for a lens shape will be a hyperbola.

He essentially derives this answer by writing the optical path length from $F_1$ to $A$, then $A$ to $D$, and putting the condition that the sum of this path length must be constant, from which we get this equation.

I can hand derive and confirm that this equation leads to a hyperbola. However, I believe the main property of a hyperbola is that the difference between the distance of a point from $F_1$ and $F_2$ is always constant, taking the value of $2a$. I'm confused how the above equation and this standard definition of hyperbola correspond.

A hyperbola can also be described by the directrix property.
If $H$ is the projection of $A$ on the directrix relative to $F_2$, we have:
$$
F_2A=eAH,
$$
where $e=n_t/n_i$ is the eccentricity of the hyperbola.
Combining that with
$$
F_1A-F_2A=2a,
$$
where $2a=F_1F_2/e$ is a constant, we get:
$$
F_1A=eAH+2a.
$$
If now $DD'$ is a line parallel to the directrix,
if $A$ lies between $DD'$ and the directrix
and $D$ is the projection of $A$ on $DD'$,
we have $AH=DH-AD$.
The above equation can then be rewritten as:
$$
F_1A=e(DH-AD)+2a=eDH-eAD+2a.
$$
But $eDH+2a$ is a constant not depending
on the position of $A$, because $DH$
is the fixed distance between line $DD'$ and the directrix.
Hence we obtain the stated result:
$$
F_1A+eAD=eDH+2a=\text{constant}.
$$

