Sometimes in Mathematics one uses the same name for two different things or two different definitions which are not related whatsoever.
But here this is not the case. I am sure you know what linear means in terms of maps between vector spaces. So if $V$ is a vector space and $A:V\rightarrow V$ is a linear map, then it satisfies $A(\alpha \mathbf v + \beta \mathbf w) = \alpha A(\mathbf v) + \beta A(\mathbf w)$.
Now something like $d/dx$ can be viewed as a map on a vector space. There are actually many different vector spaces that would work, but I suggest you just think of the vector space given by all smooth functions from $\mathbb R \rightarrow \mathbb R$. This is a vector space over $\mathbb R$ and $dy/dx$ is a linear map on it. But an expression as $(d/dx)^2$ is not linear on this vector space. Recall that $(dy/dx)^2$ acts as follows:
(d/dx)^2 (f) = (d/dx)(f) (d/dx)(f) = f' f'
Note that something like $a_n(t) d^n/dx^n + \dots + a_1(t) d/dx + a_0(t)$ is also a linear map on this vector space (if al the a_i(t) are smooth, otherwise you need to change the vector space a bit). So one way to figure out wether a ODE is linear or not is to try to find out if it acts linear on smooth functions.