Biden changed things up with his greeting, but there is no need for alarm.
This July 2023 meeting between the world leaders was their first face-to-face conference since King Charles III became king.
In the opening remarks of the meeting, strong solidarity between the U.S. and the U.K. was shown.

And I know we’ll want to do everything we can to strengthen Euro-Atlantic security."
Biden added, “We met in San Diego, Belfast, Hiroshima, Washington, here.
And couldn’t be meeting with a closer friend and a greater ally.

And I look forward to our discussions” (viaThe White House).
In 2009, former First Lady Michelle Obama gave Queen Elizabeth II a side-hug after shaking her hand.
Sometimes, following traditional customs when face-to-face with a royal causes an awkward situation.

While backstage, the Queen walked up and stood near where Mendes was waiting to go onstage.