Definition of the inverse of the Ackermann function
The Ackermann function $n \mapsto A(n,n)$ is definitely very far from being bijective, otherwise it could not grow so quickly. However, in the contexts it is usually employed (complexity theory, oh notations, ...) it is not really relevant which are the actual values of $\alpha$ point by point. You can add one or multiply by three and the results are the same, so nobody really cares about what is the actual definition. Any of your two will work.