Is greater relative AA battery capacity at high currents indicative of greater capacity at low currents?
This is a great question and I'm sorry it wasn't answered sooner. I'll give the caveat that I am not a card-carrying electrochemist, and I'm definitely not a battery researcher, even though I have worked with them in the past. I'll give my best effort.

As you note, in principal, the capacity of a battery is tracked with units of mA$\cdot$hr. In theory, this should be constant no matter the current draw.

That doesn't mean the capacity is uniform across a particular size/shape designation. The "AA" type specifies a particular shape/volume dimension, but different chemistry will give different electrochemical capacities. For example, I have plastic shells that you put a AA battery to fit "C" or "D" types - the downside is less electrochemical capacity (~2500 mA$\cdot$hr for a typical AA vs. ~8000 mA$\cdot$hr for a C cell).

One part of the electrochemical capacity is how much of the volume is filled with electro-active materials, rather than electrolyte, fillers designed to dissipate heat, or for other safety margins.

In practice, as seen in the showdown, nothing ever matches the theoretical capacity, and different batteries and different chemistries yield different practical capacities depending on the current draw.


Results @ 200 mA
Results @ 1000 mA


We see that the lithium cells get ~3000 mA$\cdot$hr under either current. The alkaline cells get ~2500 mA$\cdot$hr at low current, and much less under high current ~1500 mA$\cdot$hr.

So what's the difference?

First, as we mentioned, not every battery has the same capacity, even of the same battery type (size/shape/volume). Assume for the sake of argument that we have two 3000 mA$\cdot$hr batteries: one lithium (of some sort) and one alkaline.

Above a certain current threshold, lower currents will let batteries get closer to their theoretical capacities.

The problem is actually simple. Besides electrons, in every battery ions move through an electrolyte.


Alkaline cells use mobile $\ce{OH-}$ ions.
Lithium cells use mobile $\ce{Li+}$ ions.


At the end of the day, discharge requires ionic conductivity to match the electrical current. So which is likely to have a higher ionic conductivity? Well, $\ce{Li+}$ is smaller and usually has higher ionic conductivity.

In principal, at low current, an alkaline cell with the same capacity rating as a lithium cell, say 3000 mA$\cdot$hr should have about the same capacity. I'm not sure I've ever seen an alkaline AA battery with such high capacity -- they usually come around 2500-2700 mA$\cdot$hr theoretical capacity.

Finally, you ask if manufacturers "tune" the batteries. Put simply, yes. Besides safety considerations, different cells add compounds to minimize self-discharge, corrosion, shelf-life, etc. These can affect the amount of electrochemical capacity.

UPDATE

I realized I should clarify the very low current comment. If the current is low enough, then the rate of ionic mobility will be high enough for the chemical part of a battery to keep up with the electrical draw. So decreasing current draw will improve the capacity, up to this point - and then the real capacity should remain the same no matter how low the current draw.