I think there is some language misunderstanding going on. Likelihood function is not a probability density. Likelihood function evaluated for D is equal to prob

I think there is some language misunderstanding going on. Likelihood function is not a probability density. Likelihood function evaluated for D is equal to probability density of D (by definition). In other words, f(x;theta) as a function of x is a probability density function. f(x;theta) as a function of theta is a likelihood function. But f(x;theta) for given x and theta is just a value, which, one can say, is both likelihood and probability density.