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We call the integral of a probability density function (PDF) a cumulative distribution function (CDF).

But what's the cumulative sum of a probability mass function (PMF) called? I've never heard the term "cumulative mass function" before, and the Wikipedia page for it redirects to the CDF page, so I'm confused what the proper terminology is.

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The proper terminology is Cumulative Distribution Function, (CDF). The CDF is defined as $$F_X(x) = \mathrm{P}\{X \leq x\}.$$ With this definition, the nature of the random variable $X$ is irrelevant: continuous, discrete, or hybrids all have the same definition. As you note, for a discrete random variable the CDF has a very different appearance than for a continuous random variable. In the first case, it is a step function; in the second it is a continuous function.

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    $\begingroup$ Just to supplement the answer: if you're looking for the CDF analog of the PMF of a sample rather than of a population, then the term you're looking for is empirical distribution function. $\endgroup$
    – user541686
    Commented Sep 26, 2019 at 22:49
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I think "cumulative mass function" is correct, but it hasn't been widely adopted just yet. It makes sense to me as a more specific cumulative distribution function, a sibling to probability mass functions, similar to how cumulative density functions relate to probability density functions.

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