A simple problem. What am I doing wrong?

The problem with your computation in #3 is that you didn't account for the possibility that the first card was the king of hearts.

In the third way, the formula is ($A-$ ace of hearts, $B-$ king of hearts): $$P(A\cup B)=1-P(A^C\cap B^C)=1-\frac{51}{52}\cdot \color{red}{\frac{50}{51}},$$ however, the event $B^C$ depends on the event $A^C$. In other words, you assumed the first card is not king of hearts, however: $$P(B_2^C|B_1)=1; P(B_2^C|B_1^C)=\frac{1}{51}.$$

Probability tree diagram ($A'=A^C$):

$\hspace{1cm}$

$$B\cup B'=S; A'\cap S=A';\\ P(A'\cap B')=P(\color{blue}{(A'\cap (B\cup B'))}\cap \color{green}{B'})=\\ P(\color{blue}{([A'\cap B]\cup [A'\cap B'])}\cap \color{green}{B'})=\\ P(\{\color{blue}{[A'\cap B]}\cap \color{green}{B'}\}\cup \{\color{blue}{[A'\cap B'])}\cap \color{green}{B'}\})=\\ P(\color{blue}{[A'\cap B]}\cap \color{green}{B'})+P(\color{blue}{[A'\cap B'])}\cap \color{green}{B'})=\\ \frac{1}{52}\cdot \frac{51}{51}+\frac{50}{52}\cdot \frac{50}{51}.$$ Note: The events in blue color are the first card, while in green color are the second card. They can be differentiated by relevant indices (subscripts) as labeled by Graham Kemp in his comment.

Hence: $$\begin{align}P(A\cup B)&=1-P(A'\cap B')=\\ &=1-\left(\frac{1}{52}\cdot \frac{51}{51}+\frac{50}{52}\cdot \frac{50}{51}\right)=\\ &=\frac{101}{51\cdot 52}.\end{align}$$