Action potentials are initiated where there is a high density of voltage-gated $Na^+$ channels and that may vary from cell to cell, as you guessed. For an illustrative example see the figure below, reproduced with permission from Kole et al. (2008).
Density of voltage-gated $Na^+$ channels and $Na^+$ influx along the axon of a cortical pyramidal neuron.
Reproduced with permission © Nature Publishing Group
As you can see, in this cell type voltage-gated $Na^+$ channels, labelled in green in the left-most image, are predominantly expressed away from the axon hillock. We can also observe that the expression of voltage-gated $Na^+$ channels correlates with the degree of $Na^+$ influx in a monotonic relationship (right-most panel).
In smaller cells action potential initiation can happen closer to the axon hillock and in cells of some invertebrates even in multiple locations (Kole and Stuart 2012). What may be the advantage of using the axon initial segment (AIS) as the sole site of initiation of action potentials (APs)? Here is a direct quote from Kole and Stuart (2012):
[The AIS] has a small local
capacitance ($C$) and therefore requires less inward current ($I$),
that is, a smaller number of $Na^+$ channels per unit area, to
generate APs compared to larger structures, such as the soma
or proximal dendrites. Hence, the AIS is also an energetically
favorable site for AP initiation. Furthermore, the small capacitance
of the AIS favors rapid changes in membrane potential,
as occurs during the upstroke of the AP ($dV/dt = I/C$). Finally, it
is worth noting that having a single site of AP generation provides
neurons a single locus where inhibition can gate AP initiation. [...] Initiation of APs
further from the soma, taking advantage of the electrical isolation
of this region, is a strategy used in some neurons to increase their
capacity to discriminate the arrival time of different synaptic
M. H. P. Kole et al., Action potential generation requires a high sodium channel density in the axon initial segment. Nat. Neurosci. 11, 178–186 (2008). doi: 10.1038/nn2040
M. H. P. Kole, G. J. Stuart, Signal Processing in the Axon Initial Segment. Neuron. 73, 235–247 (2012). doi: 10.1016/j.neuron.2012.01.007