This is incorrect. Because you did not intentionally withdraw a card, and, instead, you randomly withdrew it, you do not have one less card to consider. It may seem that way, because you took out a card, and now you're' looking at it, but due to the random nature of the withdrawal of the card, all cards still need to be considered, even when the deck only has 51 cards. To be explicit: this is because while the deck only has 51 cards, it can have any combination of 51 cards, from a pool of 52 cards, so the 51 cards will not always be the same 51 cards.

The OVERALL probability, from the beginning, that the card of interest is the second one in the pile is 1/52, that's correct. But the probability that it is the second once you know that it isn't the first is different, it's a question of conditional probability (see my other comment).