I am puzzled as to what it is you two think determines the voluntary behavior of sentient animals, if not cognition. I'm not considering behavior and cognition as isomorphic -- a Venus Flytrap closing on a bug or a human speeding up his heart rate when in pain are evidently not caused by cognition, and there may be all sorts of cognitive differences that aren't reflected in behavior. But how is it possible for lions and lionesses to have systematically different behavioral tendencies without first having systematic cognitive differences to bring them about? What, are lioness's different decisions in the same situations a purely mechanical reaction to not being as heavy as lions? On its face you two appear to be disputing that this is a cause-and-effect universe. Or perhaps you two mean something different by "cognition" from what I mean.
I'm struggling for the terminology to get this out, so bear with me.
There's a difference between cognition and behavior, and you reference it somewhat. Cognition is the process by which we acquire and expand knowledge; behavior is the mechanism of response to our environment. Several of those terms in there are being used very broadly, but to narrow the scope would require pages and a degree I don't hold.
You ask whether a lioness's decisions are based purely on them not being as heavy as lions. To this, I would ask how much of a lioness's observed actions do you believe are decisions consciously made, as opposed to behaviors subconsciously acted upon? Do you think a lioness hides her cups because she's consciously considering the likelihood that the pride lion might kill them? Or do you think she hides them because her innate instincts developed over millennia drive her to that behavior?
Behavior is how we respond, and the majority of it is not conscious. The fact that it's not conscious is why we have an entire field of cognitive behavioral therapy - to bring those unconscious responses into conscious consideration so that we can alter them. Cognition is much more volitional, and includes the act of considering consequences and implications, and actually making decisions when faced with multiple potential paths.
As far as "females form thoughts through a different mechanism than that used by males" goes, yes, of course they do. Everybody forms thoughts through a different mechanism from that used by everybody else. I form thoughts through a different mechanism from that used by you two: I form them using my brain and you form them using your brains, and my brain and your brains are three different mechanisms. No people's brains are alike. When two people choose differently in the same situation, what cause is there in the universe to account for that, apart from some difference in the thought-forming mechanisms their respective brains implement?
This is splitting hairs. Sure, nobody's brains are alike... but neither is anybody's knee joint exactly the same as someone else's. And yet, all of our knees function by the same mechanics (assuming they function, of course). Some people can run and jump and their knees have no complaints. Other people's knees are just not as cooperative when it comes to athletics. Some people have high manual dexterity and can play piano beautifully, other people have low manual dexterity and struggle to feed themselves with a fork - but all of those hands have the same mechanics involved.
Similarly, the processes by which we think and learn are largely the same. No two people's brains are identical, but the mechanisms by which our brains function are the same. We all have an occipital and a temporal lobe, we all have frontal cortexes, and those sections perform the same processes. The outcomes will differ for each person, sure. And some may have better executive function where others struggle to prioritize the simplest tasks.
So with that in mind... are you and jokodo of the opinion that the brains of female humans function in a different manner than the brains of male humans? Do you think our brains have vastly different structures, or that the process by which we acquire and expand our knowledge is fundamentally different?
You could potentially argue that males and females urinate through a slightly different process. We both have urethras that attach to our kidneys, and form an evacuation path for urine, but those urethras pass through different internal structures in a way that can alter the process. I mean, crass as it sounds, women don't generally have any trouble peeing while aroused because there's nothing constricting our urethras. But even then, the majority of the mechanisms involved in the evacuation of liquid waste are much more alike than different.
When you and jokodo opine that men and women have different cognitive processes, to me you're not saying that men and women have different behavioral tendencies. You're not even saying that we have different mental strengths and weaknesses on average. To me, you're claiming that the functional processes of our brains are fundamentally different from one another. And I don't think that view is supported.
So I'll circle back around to my first comment in there: Explain what you mean by "biologically female cognition".