First, the Cambridge Dictionary definition for thankful that you quote is correct, but not complete. Merriam Webster defines thankful as
glad that something has happened or not happened, that something or
According to this definition, Facebook's use of thankful on Mother's Day makes sense.
Moving on to your question
Is there an important difference [between thankful and grateful] which everyone agrees on
Etymoline explains a difference, and, as a native English speaker its explanation makes sense to me. Whether "everyone" would agree, I can't say.
grateful: 1550s, "pleasing to the mind," also "full of gratitude,
disposed to repay favors bestowed," from obsolete adjective grate
"agreeable, pleasant," from Latin gratus "pleasing" .........
Grateful often expresses the feeling and the readiness to manifest the
feeling by acts, even a long time after the rendering of the favor;
thankful refers rather to the immediate acknowledgment of the favor by
words. [Century Dictionary] (emphasis added)
Going back to the Mother's Day example: You are thankful that you have a loving mother, but you are grateful that she gave you an interest-free loan to start your business, and you are cutting all your discretionary expenses so as to repay her as soon as you can.