May I please know the right way of saying - when you are holding an open umbrella to protect you from rain: holding an umbrella or holding up an umbrella?

Thank you very much. Merry Christmas!
"What can come down the chimney down but can't go up the chimney up?"

I would simply say "holding an umbrella." If you wish to avoid ambiguity, then say it as you did in your introduction: "holding/carrying an open umbrella."

Although "Please put up the umbrella" may mean "Please open it," it may also mean, "Please put it away." It would be simpler if it were a beach umbrella, because "putting it up" would intail both opening it and erecting it.

"Please hold up the umbrella" could apply to one which is opened or one which is closed. It's very unlikely that the speaker would mean "Please open it."

If two people are sharing the umbrella, the one who is not holding it may complain that the other isn't holding it properly. "Please hold it up!"

It's not a simple thing to pin down. "He was walking with an umbrella held above his head." "He was carrying an umbrella in the pouring rain." (We assume it's deployed and held up.) "He was carrying an umbrella" could mean under his arm.

I think you have to rely on the context to help you out. If you're telling a story about a rain storm, that will make it easier.

You could use either, although 'holding up' is more explicit.

'Holding an umbrella' could also mean 'carrying an umbrella', without opening it up/out.

Merry Christmas to you too!
Teachers: We supply a list of EFL job vacancies
Thank you very much sir.
(as requested)

The difference between the two is simply that the latter tells us that the umbrella is in an upright position.
"Holding the umbrella up"
as opposed to holding the umbrella in some oher position (down, sideways... whatever)

"Damn it Wendy! Hold the umbrella up! I'm getting wet!. Stop dragging it on the footpath!
Stop holding it down near the gutter. That might keep the ants dry, but I'm getting soaked!"
Many, many thanks for your assistance yizhivika!
Students: We have free audio pronunciation exercises.
Thank you very much too for your reply, Mike in Japan!
1. Do

-I put up my umbrella
-I opened my umbrella

mean the same?

2. What is the difference between

furled umbrella?

Do they both mean closed?
OttoJ1. Do-I put up my umbrella&-I opened my umbrellamean the same?
Yes, but 'put up' can also mean 'store' or 'wield as a weapon'.
OttoJ2. What is the difference betweenrolled&furled umbrella?Do they both mean closed?
Students: Are you brave enough to let our tutors analyse your pronunciation?