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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?
Isn't that just the difference between weight and mass?
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.
That is exactly the difference between weight and mass. And yes, that would be weightless and objects like balloons have negative weight until they reach an altitude where they're neutrally bouyant and then are weightless.
https://en.m.wikipedia.org/wiki/Weight
Weight is defined as a force and has a direction. That's what causes the movement. If the weight didn't change there would be no movement.