What does the term "undefined" actually mean?
Solution 1:
Saying that 1 divided by 0 is undefined, does not mean that you can carry out the division and that the result is some strange entity with the property “undefined”, but simply that dividing 1 by 0 has no defined meaning. That is just like when you ask whether the number 1.9 is odd or even: That is not defined. Or when you ask what colour the number 7 has.
Solution 2:
To put matters straight: Division is a function $$q:\quad{\mathbb R}\times{\mathbb R}^*, \qquad(a,b)\mapsto q(a,b)=:{a\over b}\ ,$$ whereby $q(a,b)$ is the unique number $x\in{\mathbb R}$ such that $b \>x=a$.
When we say that $\displaystyle{a\over0}$ is undefined then this means no more and no less than that the pair $(a,0)$ is not in the domain of the function $q(\cdot,\cdot)$.
Now to your three ways of understanding "undefined" in the realm of division by $0$:
If $\displaystyle{a\over0}$ could be any number, say $=13$, then this would enforce $13\cdot0=a$, which is wrong when $a\ne0$.
This is even worse. Why should $\displaystyle{7\over0}$ be the Eiffel tower?
There are circumstances where division by zero makes sense, e.g. in connection with maps of the Riemann sphere, or with meromorphic functions. There one has $\infty$ as an additional point in the universe of discourse. But these circumstances require special exception handling measures, and the "usual rules of algebra" are not valid when dealing with $\infty$.
Solution 3:
What does the term "undefined" actually mean?
In light of the already great answers provided by Carsten and Christian, I thought a more linguistic analysis of "undefined" may be in order. The following two terms are explained in the book Origins of Mathematical Words by Anthony Lo Bello:
indeterminate$\quad$ The Latin noun terminus means the end of something. From it was formed the denominative verb termino, terminare, terminavi, terminatus meaning to set bounds to. The addition of the prefix de- emphasizes that the separation is from something else and produced the compound verb determino, determinare, determinavi, determinatus meaning to fix the limits of. The addition of the negating prefix in- to the past participle of this verb resulted in the Latin adjective indeterminatus meaning undefined, unlimited.
Now to the analysis of the word originally in question:
undefined $\quad$ The Germanic negative prefix un- has been added to the word defined of Latin origin to produce the hybrid undefined. It would have been better to say indefined as we say indefinite, but it is too late now. Defined is from the verb definio, definere, definivi, definitus, which means to set the boundaries. The plural noun finus in Latin means enclosed area, territory. The force of the prefix de- is to add the sense of thoroughness to the action.
Both of these terms, especially the mentioning of "to set the boundaries" in the analysis of the term undefined, should make Christian's answer even more lucid, especially his response to your third way of understanding "undefined."
Solution 4:
From the algebraic point of view, the Real numbers form a field under multiplication and addition. If we look at the Reals as a field, there is no separate operation of "division", instead we multiply by the reciprocal. Since $0$ has no reciprocal in $\mathbb{R}$ (in fact the additive identity in a field never does), there exists no element available to multiply by to perform what would be commonly called "division by zero".
We know $0$ has no reciprocal in $\mathbb{R}$ because there is no real number $z$ that makes the following equation true: $$0z=1$$
If there were such a $z$, then we could multiply by that number and obtain an answer for "dividing by zero", but since there is no $z$, we have no way to "divide by zero".
To expand on user38858's answer, it would be like "dividing by orange". "Orange" is not a Real number, so the idea that we could perform any mathmatics with "Orange" in the context of Real numbers makes no sense. There is no real number that represent the reciprocal of $0$, so it also makes no sense to attempt to perform any mathematics on such an object.
From the algebraic standpoint, I would say that "Division by zero does not make sense" is the closest of your three concepts. Maybe I would change it slightly to say "Division by zero is not even possible".