The article is correct. The nit from grandparent is, that it uses some outdated language. So if you do not look to closely in special relativity, you will find that the mass quite often appears together with the Lorentz factor, which describes time dilation. This lead historically to the claim that moving objects are heavier than the same object at rest. But since this does not hold in general relativity, it is nowadays usually assumed that mass is always the rest mass. And so the article reads a bit outdated.
Start off with this: relativity is complicated, and unintuitive from the standpoint of human experiences.
So, let's talk about gravity. Gravity is a force between objects which have a certain kind of property, let's call it property X. Gravity has a precise quantitative relationship with property X, the more property X an object has the more gravitational force it exerts.
What is "property X"? It's energy. And here is where things get a bit complex, because most people would instead have said that "mass" is property X. The problem with that is mass becomes variable depending on the reference frame, and it turns out to add a lot of excessive complexity to discussing things, especially when precision is required.
So you could imagine talking about mass as the equivalent of energy, which is typically an accurate viewpoint, and then you get to the idea of "relativistic mass". Which is the adjusted "property X" value of an object which might be traveling at relativistic speeds in a given reference frame.
Relativistic mass, or property X, can be a helpful mental model in some ways, and in normal uses of English it's often a more useful way of thinking about things. But it's also problematic because it's ambiguous.
This has led to a bit of an impedance mismatch between the way physicists talk about relativistic effects and the ways that it's more natural to talk about such things in plain English. In English "property X" is mass, but in physics it's actually energy, and it's difficult to get people to fully grok the intimate relationship between energy and mass.
Physically, mass is just a special name for invariant, or rest, energy, the energy of an object in the reference frame where the object is stationary. It's all energy, but it's important to separate out rest-energy vs. energy in a given reference frame, and so forth.