climb or climb up? GRAMMAR QUESTION

I read in grammar book for usage part that states do not use "climb up" because the action of climbing is known as up ,so just use "climb" instead. However, I still see in some text that the author writes "climb up" so I wonder which way is correct and how do we know the different between climb up and climb down if we are not using "climb up". The grammar book that I read doesn't mention about "climb down".

I'm by no means a grammatician, but I do know I have also read authors use the term, "Climb Up", but I haven't ever heard anybody say "climb down", other than this, I don't know what to say except I would stick with just "climb".

You get to use prepositions in your own way. Do what you like.
I didn't want to climb that hill.
I didn't want to climb the mountain.
I had to climb back down the mountain.
Then, I had to climb up the mountain again.
First I climb up, and then I climb down.

In most sentences, you can just write climb.
My mother climbed into the ring and knocked out Mike Tyson.
Mike Tyson woke up and climbed back to his feet.
My mother does not like to climb stairs.

In my view, i always use climb in many sentences
For example, I am on vacation with my friends to climb a mountain to see a sunset.

I think up or down tells you about actions you refer to.
For example, i want to climb down a tree because it is very dangerous and climb up to breathe breezy wind

This is my examples for you and you can know more by looking up Oxford dictionary.

i think both are correct? as normally we just use climb (everyone knows that climb in upward direction).
For me, i will always use climb for upward, and climb down for opposite direction.

Thanks for all the helps! I'm clear now