OR in real life vs OR in Mathematical Logic
Both of the English sentences you've written are correct - the word "or" in English is ambiguous, it can mean both the logical operation OR and the logical operation XOR depending on the context.
In mathematics, we have decided that "or" will only mean the logical operation OR.
It is usually the case that in mathematics, it is ideal for things to resemble our everyday experience as much as possible (e.g., 3+2=5 resembles our experience with three apples and two apples together making five apples). However, in mathematics we also usually want more precision than everyday experience, at least for something as basic as what the word "or" means. There is no requirement that any mathematical term resemble the corresponding thing in English, or anything else from the real world, but we do generally strive for the mathematical terms we create to be somewhat evocative of the appropriate ideas.
The English word "or" has lots of different meanings, and the logical OR operation is only one of those meanings.
Here are some sentences that seem to use the word "or" to indicate the logical OR operation:
Either David or Nick will be at home.
I hear a hissing noise. Either the air is leaking out, or there are snakes in here.
Sometimes the word "or" indicates the logical XOR operation:
I see that each person is only taking a slice of cake or a slice of pie. Why not both?
The word "or" can be used to indicate a list of options, in which case the meaning is neither the logical OR nor the logical XOR. In fact, the meaning is more similar to a logical AND.
You can have the soup or a salad. (You are allowed to choose the soup, and you are allowed to choose the salad, but you are not allowed to choose both.)
This car has two color options: blue, or red. (Blue is one of the color options, and red is one of the color options.)
The blue car costs \$20,000 and the red car costs \$30,000. I have \$40,000 to spend, so I can afford either the blue car or the red car. (I can afford the blue car, and I can afford the red car, but I can't afford both.)
The word "or" is used to ask "which one?" In this usage, the word "or" definitely doesn't indicate any logical operator, because it doesn't form a yes-or-no question:
Would you like cream or lemon, Mr. Feynman?
The word "or" can be used to make a threat:
Give me some ice cream, or I'll scream!
All of these are different meanings of the word "or", and only the first meaning corresponds to the mathematical "or".
Do operations like OR, AND in mathematical logic need to resemble OR and AND from English language? and what are consequences if they don't?
Well, the logical OR and AND operators are named after the corresponding English words, because the meanings of the logical OR and AND operators resembles the meanings of "or" and "and" in English.
There doesn't have to be a correspondence. For example, the logical IF-THEN (material implication) operator doesn't correspond to any common meaning of the word "if" in English. The consequence of this is that millions of students are confused by this operator.
Here's an example of a confusing IF-THEN sentence. Using the logical IF-THEN operator, the sentence "IF there are people on Mars, THEN there are no people on Mars" is true. But using the ordinary English meaning of the word "if", the sentence "if there are people on Mars, then there are no people on Mars" is false, or at least disagreeable.
You should sparingly use the word "either" (with "or") in mathematical writing. Logically it is equivalent to plain "or" but in English use it tends to emphasize "not both." So this is just confusing for the reader who has to wonder: "we have agreed it does not change the meaning, but are you using it because you think it does?" You will also notice in plain English sometimes "or" is underlined, capitalized or written in boldface to mean "exclusive." This is another thing you shouldn't do in math writing.
In mathematics you have to resort to "exactly one of A or B" or "A or B but not both." And you should need extra verbiage: exclusive-or is a strictly stronger claim with more information. So it should take more time to express this since your proof requires more involved information.