Mindfulness is simply noticing.
I remember being perplexed by the "overthinking" thing early on in my own practice. To me, the term isn't meant to imply that I should not think at all. It simply means that I'm attaching a whole lot of unnecessary (and false) importance on my thoughts, enslaving myself to them.
Mindfulness encourages me to see my thoughts for what they are, without judgement if they're right or wrong, nor expectation that I have to follow them, or even believe them!
I read this and smiled:
"As I mentioned in another thread, I'm someone who was taught that emotions (especially *acting* on emotions) is silly, ludicrious and only permissible on the occasional exception.
So the idea of walking round a suburb + park based solely on how which street makes you FEEL today is kind of a foreign concept."
I was taught that this exact sentiment applies to thoughts and feelings both. With mindfulness, I am able to choose to act instead of react.
Plus, mindfulness does not mean feeling. It has nothing to do with emotions, except that it's a great tool for becoming aware of them.
I love in the example you gave, you noticed what your ideas and concepts about how to approach emotions in life are... so it is the act of noticing that is mindfulness. The ideas and concepts and feelings themselves are simply the things that you noticed.
It's easier for me to simply observe and notice when I stop taking what I think and feel so seriously. Feelings and thoughts come and go, like clouds in the sky. Big deal.
So while walking the dog, no matter what thoughts or feelings arise, my practice of mindfulness allows me to just observe. Nothing more. Whether the dog is sleeping peacefully or running around in mad circles is irrelevant, has nothing to do with how mindful I am (or can) be.
Not sure if any of this helps, but it sure gave me some great stuff to ponder.