Very interesting, but I don't see how replacing the same volume of air in our lungs with helium doesn't make you lighter. It's the same volume, so the volume displacement zeroes out in any equation - I think that poster may mean as compared to empty lungs. Even then I think they're mistaken - otherwise a blimp/balloon wouldn't work, as it too is displacing air around itself, and increasing in volume.
That's a fun question, and kind of depends on how you see it. If you're going by "standing on and reading a scale in Earth's atmosphere", I believe the scale would read ever so slightly less. However, this is kind of mixing up weight and how we measure weight. Helium still has mass, it's not negative mass, it still is affected by gravity and gets pulled down by it like everything else. It's just that it's less dense than other gasses in the atmosphere, and so the buoyancy overpowers gravity and it floats. So, you with deflated lungs actually weigh less than you with lungs inflated with helium, even if that's not what the scale reads!
I think your closing statement contradicts your earlier statement about weighing less (though I get the point you're trying to make about mass).
Do we consider weight a sum of all interactions in a given place (including atmosphere)? I'd say we do, since our atmosphere accounts for a notable portion of our weight, and I've never seen a scale with a negative tare to account for atmospheric weight.
Weight is purely the effect of gravity on mass. Do you consider yourself to weigh negative weight when you're floating in a pool and the scale is on the bottom of it?
Less the difference between weight and mass and more the affect of buoyancy on your method of measurement. If you float in water, it doesn't mean you're weightless in water. It just means the buoyant force of the water overpowers gravity.
You may be slightly more buoyant (and therefore apply less force on a scale) everytime you breath in. It's not the presence of air that has this effect, it's the decrease in density of your total body (mass/volume) that has that effect. (Helium just contributes a fractional more difference in density compared to air, but how much you breath in probably matters much more than what you breath)
Except, maybe not. Because the air you breath in partially dissolves in your blood. Dissolved matter does not decrease density, rather the opposite: it packs tightly into the voids, increasing mass for the same volume.
How much of an effect this has is hugely debatable, probably depends on a dozen biological and circumstantial factors, and this is where my knowledge ends. But it's fun to imagine.
However, if you can imagine inhaling but holding your breath at the same time, creating a vacuum in your lungs, then yes, you would be more buoyant, even more than inhaling helium, and the scale would read slightly less.
You are adding more mass to your body, but you are also decreasing your body's density. So this should in fact make you lighter on the surface of the earth.