Is there any difference between the phrases "I lifted my left hand" and "I raised my left hand"?
When talking about hands, "raise" almost always refers to holding it in a position where the palm is facing forward at or above the height of your head.
This is commonly used in swearing oaths and when asking or answering questions in a classroom.
Raise your hand and repeat after me.
"Lift" has no standard usage with hands so saying "lift your hand" would simply mean, "pick it up off of whatever it's sitting on". It doesn't imply any particular position other than higher than where it was.
Lift your hand off the table so I can get the tablecloth out from under it.
AFAIK, lift can only be used in a literal sense, i.e. in the physical context of an object going from point A to a higher point B.
In aeronautics, this is also the name of the force that makes a plane or helicopter stay aloft.
Raise, on the other hand, can also be used when talking about other things that may "go up", e.g. a salary raise.
Please lift your feet, so I can sweep the rug!
Overjoyed to finally see his girlfriend back after missing her for so long, he took her in his arms, lifted her up and spun around.
At the dinner table, she raised the issue of who was finally going to do the dishes once more.
The rollercoaster rises six times, the highest point being at 18 meters.
I have always thought that raise is meant to happen under the power of the subject. Lift is meant to happen by the power of a third-party. Both are in the same direction of up. As far as hands go, A person can raise their hand as that hand is a part of them. If speaking about a person's anatomy, "he lifted his hand" would be more appropriate.