Is the common saying, "Start on the right, even if Omar is on the left," correct, and does handshaking begin from the right, especially with men present at the head of gatherings?

Starting with the right is more preferable, due to what was narrated by Aisha, may Allah be pleased with her, that the Prophet, peace and blessings be upon him, loved to start with the right in all his affairs. And in the story of the Prophet, peace and blessings be upon him, drinking and offering it to the young boy on his right before the elders.