It may be the case that any two surfaces equivalent by the bijection definition are also deformable into each other given sufficiently many dimensions. Certainly I can't immediately think of a counterexample...
I was wrong about the internally knotted torus, incidentally. The book I mentioned does contain diagrams for untying a knot in the hole of a torus, but only in the case where the torus has two holes and the unknotted one goes through the middle of the knot in the other. An internally knotted one-hole torus can't be untied (in three dimensions) any more than an externally knotted one; as a simple proof, consider a line drawn along the surface such that it goes through the knotted hole, loops round the outside of the torus and joins back up with itself. This closed curve is tied in a simple overhand knot, and hence no continuous deformation of the torus can transform it into an unknotted curve. (I feel silly for not having spotted that to start with.)
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I was wrong about the internally knotted torus, incidentally. The book I mentioned does contain diagrams for untying a knot in the hole of a torus, but only in the case where the torus has two holes and the unknotted one goes through the middle of the knot in the other. An internally knotted one-hole torus can't be untied (in three dimensions) any more than an externally knotted one; as a simple proof, consider a line drawn along the surface such that it goes through the knotted hole, loops round the outside of the torus and joins back up with itself. This closed curve is tied in a simple overhand knot, and hence no continuous deformation of the torus can transform it into an unknotted curve. (I feel silly for not having spotted that to start with.)