This is how I introduce myself every visit. When you offer your hand to a cat to sniff it is giving them a moment to pick up your scent and adapt to having you now in the home.
What you should never do is enter their home and scoop them up immediately. Some cats may like this but most will not. They need a moment to get used to you in their space.
If a cat is outside when you approach it, do not bend down and pick it up. Crouch down to the cat's level and offer your hand first. You will know immediately if the cat wants you to invade it's space. If the cat backs off, do not scoop it up. If it approaches with tail up and looks at you then you can safely pick it up.
When Mr. Ed first wakes up I do not move him even if it is time to make up the bed. He is a very old cat (21 years) and he needs a bit of time to stretch and shake the cobwebs from his triangular head before he jumps off the bed. This is the look I get when I say to him, "Ed, it's time to wake up now and have breakfast".
I'm off to bunny-sit 2 bunnies now. They are not easy to photograph as they are in cages but I will try to get one as I cuddle them. I'll bring them some lettuce for a Saturday night treat.
Enjoy your Saturday!