There are two basic results from probability that are at work in Bayes' theorem. One is a way of rewriting a joint probability density function:
$$p(x,\,y)=p(x\,|\,y)p(y).$$
The other is a formula for computing a conditional probability density function:
$$p(y\,|\,x)=\frac{p(x,\,y)}{p(x)}.$$
Bayes' theorem just stitches these two things together:
$$p(\theta\,|\,x)=\frac{p(x,\,\theta)}{p(x)}=\frac{p(x\,|\,\theta)p(\theta)}{p(x)}$$
So both the data $x$ and the parameters $\theta$ are random variables with joint pdf
$$p(x,\,\theta)=p(x\,|\,\theta)p(\theta),$$ and that's what shows up in the numerator in Bayes' theorem. So writing the likelihood as a conditional probability density instead of as a function $L$ of the parameters makes clear the basic probability at play.
That all said, you'll see people use either, like here or here.