If forced to choose one, which degree would be better for someone that wants to work directly with artifacts?

That depends- what do you want to do with artifacts? Preserve them? Display them? Catalog them? Analyze them? Taste test them? Trace their provenance?

The terms "artifacts studies" and "material culture" are usually actually associated with archaeology. If you are interested in studying artifacts and using them to learn about their makers' culture, I'd encourage you to take a look at an archaeology or anthropology degree. "Material culture" is basically the two word definition of archaeology. Though the typical image of an archaeologist is an excavator laboring away in the dirt square-jawed, with perfect facial stubble and their white shirt unbuttoned exposing their sweat-covered pecs a la Harrison Ford in the Temple of Doom, much, if not most of the sweating in archaeology is done in the lab. Would that mean you might have to pick up a trowel and get on your knees a couple days a year? Probably. But if you are interested in studying artifacts, archaeology is the way to go.

That's not to say Museum Studies is not archaeology. Museum specialists and archaeologists interact, crossover, and support each other all the time. Some Museum Studies programs will focus on artifact work. It's not typically research on them though, especially in a broader "material culture" context. Most programs emphasize preservation and restoration, displaying the pieces, designing exhibits, and managing collections. At my school, the Museum Studies program is in the Arts department. Heck, some Museum Studies programs are in the Information Sciences department.

Most Cultural Heritage programs will focus more on the education, management, and preservation side of things: setting up and running National/State/Local Historic Sites, restoring buildings and landscapes, managing interpretive programs and tours, creating public history curricula, etc. This is not going to be the experience you're looking for.

Both of these degrees will train you much more for management or public education than they will for research. Before you make any decision, make sure to look at the classes offered/required for programs. That will be more much more important than the name of the degree you get. What really matters is your work/research/field experience. That will be the greater determining factor in what you end up doing.

With regards to this specific question, I would direct you towards Museum Studies. At the same time, consider an archaeology MA with specialization in artifact analysis.

Actually, there are several programs around the world that specialize in Material Culture! Many Material Culture Programs are closely associated with ALL sorts of museums so you get to work closely with objects, curators, anthropologists, and conservators daily. If you haven't looked into any of those programs, I'd recommend it. Otherwise, Cultural Heritage Management sounds like the way to go.

Based on your interests, I would strongly advise against Museum Studies, because that focuses more on "museum management..." Working in development offices, reception, etc. Sounds like you want something more hands-on and research-based. Material Studies Programs are also often geared towards museum work, so if you aren't interested in academia that might be a great path to explore.