What's the reason for the second condition of the pumping lemma(s)?
Consider the language of words over $\{a,b\}$ consisting of words with an equal number of $a$s and $b$s. This language is context-free but not regular. To show that it is not regular using the pumping lemma, you start with the word $a^pb^p$ and pump $y$, which must consist solely of $a$s. Without condition (2), this wouldn't work: you can pump $ab$ (in the middle) and remain inside the language. Moreover, for any word that you start with, you will be able to find one of the substrings $ab,ba$ which you could then pump.

Similar considerations show that condition (2) is needed for proving that the language over $\{a,b,c\}$ consisting of those words having an equal number of $a$s, $b$s and $c$s is not context-free.