What does the term "undefined" actually mean?
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.
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 $\infty$.
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."