I want to compute the fundamental group of a bucket.
Wrong proof:
If I retract the bucket onto the disk at the base what I get is a disk with a handle attached to it by two points, which in turn is homotopically equivalent to a disk with a circle attached to it by a point.
Hence since the fundamental group of the disk is trivial and the fundamental group of the circle is $\mathbb{Z},$ I get that the fundamental group of the bucket is $\mathbb{Z}*\text{id}.$
Now, I already know the computation is right but the proof is wrong.
This is because the retraction of the bucket onto the disk cannot be done in this case because the bucket has the handle. The retraction would make us "lose the information about the handle".
But now consider this other cases:
If instead we imagine a "bucket without handle", this time we could retract it onto the disk to obtain that the bucket without handle has trival fundamental group and it would be a sound proof.
In a similar way, if we consider a bucket with a hole at the bottom, that is a cylinder with a handle attached to it, then in this case we can retract the cylinder onto $S^1$ without "losing information" about the handle, and we would get a bouquet of two circles which has fundamental group $\mathbb{Z}*\mathbb{Z}.$
So, the question is
Why the proofs of these two last cases - bucket without handle or cylinder with a handle - are sound while the first one is not correct? What is the difference? Where is the problem in the first proof?