Is Wolfram Alpha linear independence wrong or am I missing something?

It's because WolframAlpha interprets your input as one vector, i.e. the space of the single vector $(t, t^2+1, t^2+1-t)$.

An appropriate input would be (treating $1$, $t$ and $t^2$ as basis vectors):

linear independence (0,1,0), (1,0,1), (1,-1,1)

which outputs linearly dependent.

You can find other input examples for linear algebra here.

Another way to check for linear independence in W|A is to compute the Wronskian, say with the input "wronskian(($t$, $t^2+1$, $t^2+1-t$), $t$)", which results in $0$ so the set of functions is indeed linearly dependent.