I find that many of my students think the same way. Instead of thinking about null linear combinations, they usually prefer to think in terms of vectors as linear combinations of other vectors. And honestly, I probably do too. The definition of linear independence that is most intuitively geometric to me, is that no vector in the list can be expressed as linear combination of the others. This is equivalent to the other definitions of linear independence.
The negation of this is that some vector (not all vectors) in the list can be written as a linear combination of others. That is linear dependence. It has nothing to do with non-zero linear combinations (otherwise, as you pointed out, adding $0$ to the list will preserve linear independence). The zero vector is always a linear combination of the other vectors, adds nothing to the span, and therefore nothing to the dimension.
There are other cases, aside from $0$, where not every vector in a linearly dependent list can be expressed a a linear combination of others. For example,
$$((1, 0), (2, 0), (0, 1))$$
Some vectors (i.e. $(1, 0)$ and $(2, 0)$) can be expressed as linear combinations of the others, but not all. There is still dependency in the list.
Hope that helps.
