I have a bit trouble distinguishing the following concepts:
Are some of these interchangeable? Which of these are defined with respect to probability spaces and which with respect to random variables?
The difference between the terms "probability measure" and "probability distribution" is in some ways more of a difference in connotation of the terms rather than a difference between the things that the terms refer to. It's more about the way the terms are used.
A probability distribution or a probability measure is a function assigning probabilities to measurable subsets of some set.
When the term "probability distribution" is used, the set is often $\mathbb R$ or $\mathbb R^n$ or $\{0,1,2,3,\ldots\}$ or some other very familiar set, and the actual values of members of that set are of interest. For example, one may speak of the temperature on December 15th in Chicago over the aeons, or the income of a randomly chosen member of the population, or the particular partition of the set of animals captured and tagged, where two animals are in the same part in the partition if they are of the same species.
When the term "probability measure" is used, often nobody cares just what the set $\Omega$ is, to whose subsets probabilities are assigned, and nobody cares about the nature of the members or which member is randomly chosen on any particular occasion. But one may care about the values of some function $X$ whose domain is $\Omega$, and about the resulting probability distribution of $X$.
"Probablity mass function", on the other hand, is precisely defined. A probability mass function $f$ assigns a probabilty to each subset containing just one point, of some specified set $S$, and we always have $\sum_{s\in S} f(s)=1$. The resulting probability distribution on $S$ is a discrete distribution. Discrete distributions are precisely those that can be defined in this way by a probability mass function.
"Probability density function" is also precisely defined. A probability density function $f$ on a set $S$ is a function specifies probabilities assigned to measurable subsets $A$ of $S$ as follows: $$ \Pr(A) = \int_A f\,d\mu $$ where $\mu$ is a "measure", a function assigning non-negative numbers to measurable subsets of $A$ in a way that is "additive" (i.e. $\mu\left(A_1\cup A_2\cup A_3\cup\cdots\right) = \mu(A_1)+\mu(A_2)+\mu(A_3)+\cdots$ if every two $A_i,A_j$ are mutually exclusive). The measure $\mu$ need not be a probability measure; for example, one could have $\mu(S)=\infty\ne 1$. For example, the function $$ f(x) = \begin{cases} e^{-x} & \text{if }x>0, \\ 0 & \text{if }x<0, \end{cases} $$ is a probability density on $\mathbb R$, where the underlying measure is one for which the measure of every interval $(a,b)$ is its length $b-a$.
This is my 2 cents, though I'm not an expert:
"Probability Measure" is used in the context of a more precise, math theoretical, context. Kolmogorov in the year 1933 laid down some mathematical constructs to help better understand and handle probabilities from a mathematically rigid point of view. In a nutshell - he defined a "Probability Space" which consists of a set of events, a ($\sigma$)-algebra/field on that set ($\approx$ all the different ways you can subset that original set), and a measure which maps these subsets to a number that measures them. This became the standard way of understanding probability. This framework is important because once you start thinking about probability the way mathematicians do, you encounter all kind of edge cases and problems - which the framework can help you define or avoid.
So, I would say that people who use "Probability Measure" are either involved with deep probability issues, or are simply more math oriented by their education.
Note that a "Probability Space" precedes a "Random Variable" (also known as a "Measurable Function") - which is defined to be a function from the original space to measurable space, often real-valued. I'm not sure, but I think the main point here, is that this allows us to use more "number-oriented" math, than "space-oriented" math. We map the "space" into numbers, and now we can work more easily with it. (There's nothing to prevent us to start with a "number space", e.g., $\mathbb R$ and define the identity mapping as the Random Variable; But a lot of events are not intrinsically numbers - think of Heads or Tails, and the mapping of them into numbers 0 or 1).
Once we are in the realm of numbers (real line $\mathbb R$), we can define Probability Functions to help us characterize the behavior of these fantastic probability beasts. The main function is called the "Cumulative Distribution Function" (CDF) - it exists for all valid probability spaces and for all valid random variables, and it completely defines the behavior of the beast (unlike, say, the mean of a random variable, or the variance: you can have different probability beasts with the same mean or the same variance, and even both). It keeps tracks on how much the probability measure is distributed across the real line.
If the random variable mapping is continuous, you will also have a Probability Density Function (PDF), if it's discrete you will have a Probability Mass Function (PMF). If it's mixed, it's complicated.
I think "Probability Distribution" might mean either of these things, but I think most often it will be used in less mathematically precise as it's sort of an umbrella term - it can refer to the distribution of measure on the original space, or the distribution of measure on the real line, characterized by the CDF or PDF/PMF.
Usually, if there's no need to go deep into the math, people will stay on the level of "probability function" or "probability distribution". Though some will venture to the realms of "probability measure" without real justification except the need to be absolutely mathematically precise.
