I think you are getting confused by two allusions that happen to share the words "on the wall", but which have nothing to do with each other.
"The handwriting on the wall" is a reference to the book of Daniel in the Bible. According to this book, the imminent doom of the city of Babylon was predicted by handwriting that miraculously appeared on the wall during a drunken party, and which the prophet Daniel then interpreted to mean that Babylon would soon fall. (Some commentators say that the handwriting was not miraculous in any way but was simply ordinary graffiti, to which the king and Daniel attached prophetic significance. But whatever, not here to debate theological questions, just to explain the reference.)
Note that the reference is unlikely to be recognized without mentioning the handwriting. It's not just something on the wall, it's the handwriting on the wall. And it's not really a discussion of probability -- maybe you were just struggling for words there -- but a warning of impending disaster. No one uses this phrase to refer to something good, but only to something bad. And it is normally understood to mean not a probability but a certainty.
"Liar, liar on the wall" and "teacher, teacher on the wall" are references to the fairy tell of Snow White. In the fairy tale, the evil queen has a magic mirror. She regularly asks this mirror, "Mirror, mirror on the wall, who's the fairest one of all?" "Fairest" here meaning "prettiest". As long as the mirror assures here that she is the prettiest woman in the kingdom, she is satisfied. But when the mirror tells her that Snow White is prettier than she is, she becomes furiously jealous and orders Snow White to be killed.
You have a hunting dog that helps you catch a wild animal. If it climbs up a tree, the dog is supposed to bark at the tree it climbed up. Imagine the situation where the dog is barking at a different (wrong) tree and you would be very disappointed at the dog.
- It could mean as a dog:
You are wasting your efforts and energy by pursuing the wrong path. In
other words, you made the wrong choice.
- As a tree:
You are asking the wrong person. You have to ask somebody else. Leave me
alone. I am not the tree you want.
The verb bark means the following:
to make the characteristic short loud cry of a dog.
It has nothing to do with the noun bark:
the tough exterior covering of a woody root or stem.
[Merriam-Webster]
Best Answer
Only considering this particular sentence, the meaning is exactly the same, because the paint won't come off as a complete layer due to the nature of paint on a wall.
But consider the following sentence:
The road was wet.=> meaning that the whole road was wet of rain.The road was wet in places.=> meaning that the road was mostly dry, but some people emptied their buckets of washing water onto the road, making it wet in places.Or consider the following context:
The house wasn't renovated for fifty years, and the paint was peeling off the wall.=> most of the paint peeled off during the last 50 years.The house wasn't renovated for fifty years, and the paint was peeling off the wall in some places.-> most of the paint stayed on, despite the fact that it was 50 years old.