Your intuition for linear (in)dependence is very close. Based on your intuition, the definition you're looking for is:
$\{v_1, ..., v_k\}$ is linearly dependent if there exists an index $i$ and scalars $c_1, ..., c_k$ (excluding $c_i$) such that $v_i = \sum_{j \ne i} c_j v_j.$
You can prove that this is equivalent to the standard definition.
Notice how this differs from your proposed definition:
(1) It says there exists a $v_i$, not for all $v_i$.
(2) There is no zero restriction on the $c_i$.
(1) is important because all it takes is a single redundancy to get linear dependence. Not all vectors have to expressible in terms of the others. To see why this is the case, just think about the case where a set $\{v_1, \ldots, v_k\}$ is already dependent and then I suddenly add a $v_{k+1}$ which cannot be expressed as a linear combination of $v_1, \ldots, v_k$. Adding a vector to a dependent set shouldn't turn it into an independent set.
As for (2), the standard definition needs to say that $c$'s can't be all 0 because you don't want $\sum 0 v_i = 0$ to imply dependence. But with the above definition, you've already singled out a vector to have a coefficient of 1 (which is not 0) so you don't need any condition on the c's anymore.
