Is Morpheus in the construct that explains the matrix to Neo just a part of the construct and not the real Morpheus? The real Morpheus plugs in Neo and then is in the construct almost immediately. It seems too fast for Morpheus to plug in.
WAG: It is the “real” Morpheus. Even though they hooked Neo up first, maybe they didn’t turn it on until Morpheus was hooked up. Recall that when Neo is learning Kung Fu, etc, he is hooked up to the machine and is still conscious at the same time. So, perhaps they hooked him, and it felt a little weird, but he was still sitting around - conscious and in pain/shock - until Morpheus is hooked up.