Is it accurate to call P(D|H) the 'likelihood' in Bayesian inference? In Bayesian inference, the term $P(D|H)$ is sometimes called the likelihood, as in the example below from Olshausen (2004):

$$ P(H|D) = \frac{P(D|H)P(H)}{P(D)} $$
The term $P(D|H)$ is called the likelihood function and it assesses the probability of the observed data arising from the hypothesis...

However, in his introductory paper to the term likelihood, Etz (2018) states the following:

A critical difference between probability and likelihood is in the interpretation of what is fixed and what can vary. In the case of a conditional probability, $P(D|H)$, the hypothesis is fixed and the data are free to vary. Likelihood, however, is the opposite. The likelihood of a hypothesis, $L(H)$, is conditioned on the data, as if they are fixed while the hypothesis can vary. The distinction is subtle, so it is worth repeating: For conditional probability, the hypothesis is treated as a given, and the data are free to vary. For likelihood, the data are treated as a given, and the hypothesis varies.

In other words, Etz (2018) uses $P(D|H)$ as an example of a conditional probability that is 'the opposite' of likelihood. Assuming this is correct, why is $P(D|H)$ still often called the likelihood in Bayesian inference? Is this incorrect?

This seems to be mostly matters of convention. Let me say mine, and clarify the actual important properties besides the nomenclature.
$$
\text{Likelihood} = L(\theta) = P\left(\text{Observed data} \, \middle| \, \theta\right)
$$
Note well, that here we consider the observed data. This is neither a distribution in terms of data (since data was fixed to the observed data) or parameters, $\theta$.
$$
\text{Sampling distribution} = P\left(\text{data} \, \middle| \, \theta\right)
$$
This is a distribution of the data and could be used e.g., to draw pseudo-data. This would usually only be used frequentist inference (which violates the likelihood principle) though may be used in Bayesian inference in: approximate Bayesian computation, simulation based calibration, for computing e.g., prior predictive distributions, and lastly objective Bayesian priors. Broadly speaking though, as we condition only on the observed data in Bayesian analysis, it obeys the likelihood principle and requires only the likelihood.
