Definition of “infix” in formal languages

I've got a simple question:

Let's say we have the following definition of a language over some alphabet: $L = \{w \mid w \text{ contains the infix } aab\}$

Does that mean $aab \in L$? or does "$aab$" have to be "wrapped" on both sides by other letters?

Yes, if $A$ is the alphabet, then $L = A^*aabA^*$.
Now, what about terminology? Lothaire in the first chapter of his book Combinatorics on words gives the following definitions. A word $v \in A^*$ is said to be a factor of a word $x \in A^*$ if there exist words $u, w \in A^*$ such that $x = uvw$. A factor is proper if $v \not= x$.
In my opinion, the word infix is also perfectly fine, since it gives a natural counterpart to prefix and suffix. Terms involving "sub" like subword and substring are also frequently used in the literature. I personally prefer to follow Lothaire and say that a word $v \in A^*$ is a subword of a word $x \in A^*$ if $v = a_1a_2 \cdots a_n$ (where $a_i \in A$ and $n \geqslant 0$) and there exist $y_0, ..., y_n \in A^*$ such that $x = y_0a_1y_1a_2 \cdots a_ny_n$. This definition makes the term subword a synonym of subsequence, which I find quite natural.