What's the proper group relation in the definition of quaternion group？

Question

I was reading a book about group representation,and there is an exercise to calculate all the complex irreducible representations of the quaternion group Q defined as $\text{Q}=\langle i,j\mid i^4=j^4=1,jij^{-1}=i^{-1}\rangle$. My calculation shows that this is a group of order 16, with 10 conjugacy classes, eight 1-dimensional irreducible reps and two 2-dimensional irreducible reps. While I was stuck on the calculation of the 2-dimensional reps, I found that the answer says that there are only 8 elements and five conjugation classes, which coincide with the usual deinition of quaternion group $\text{Q}=\langle i,j\mid i^4=1,i^2=j^2,jij^{-1}=i^{-1}\rangle$ but not the one I was working on I think. So my question is that are these two definition equivalent? If so, I wonder why the former relations can lead to $i^2=j^2$. If not, then I want to know how to caculate the 2-dimensinal reps of the former 'quasi-quaternion' group.