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.

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.