I'll want to use a little math to explain some of this, but will place emphasis on conceptual understanding.
The pdf actually has a really good quote about this in the intro, and its motivation is to try to unteach this common notion of mass depending on speed. It tries to convince you that using "rest mass" and "relativistic mass" are unnecessary. It even argues that "rest mass" itself is a lousy term because that suggests there is some other mass related to motion. I agree.
This famous equation and the concept of mass increasing with velocity indoctrinate teenagers through the popular science literature, and through college text-books. According to Einstein, "common sense is a collection of prejudices acquired by age eighteen. "It is very difficult to get rid of this "common sense" later: "better untaught than ill taught." As a result one can find the term "rest mass" even in serious professional physics journals. One of the aims of this book is to help the reader to get out of the habit of using this term.
Another useful quote:
In the modem language of relativity theory there is only one mass, the Newtonian mass m, which does not vary with velocity~ hence the famous formula E = mc2 has to be taken with a large grain of salt.
Here I would disagree with the "Newtonian mass" because that raises more questions like what is Newtonian mass, and what's a non-Newtonian mass? Ignore it. The important part is it does not depend on speed. I'll also disagree with saying that E=mc2 has to be taken with salt. The formula is right! The catch is that you have to know what the m and the E mean. This E is not the kinetic energy, or the total energy. It is the internal energy, and that is why it does not depend on speed.
What does depend on speed is the total energy, which is mc2 times [math] ("gamma"). Gamma is what captures the essence of relative motion in relativity. At slow speeds it is very close to 1. At very fast speeds, approaching the speed of light, it goes to infinity.
We can also say that E=mc2 is actually a part of a more general formula, [math]. Here E is the total energy, and again mc2 is the internal energy. p is the momentum. This explains why massless particles like photons have momentum. The formula p=mv that is taught early on in physics is Newtonian. In relativity, massless particles have momentum p=E/c.
So anyway, what the heck is mass?
Mass is a measure of the internal energy of an object. It is the sum of its parts, plus an energy associated with how those parts are bound together. Every fundamental particle has an established mass. A proton has a mass of 1.67x10-27kg. Always.
That the mass is associated with the energy involved in binding the parts together is very interesting, and is the thrust of E=mc2. It means for example that a stretched spring literally weighs more than it does when relaxed, because there is an additional internal energy associated with being stretched. It is also the basis for how the Sun emits energy by fusing protons together. When you combine the protons, they reach a lower energy state than if the protons were separated. So the internal energy is little bit less than the sum of the parts, and the difference was emitted as radiant energy when they fused. Hence, sunlight!
Finally, because the value of c is big, and squaring it makes it even bigger, the relationship says that a tiny change in the mass represents a huge change in the internal energy. E=mc2 describes mass-energy equivalence, but it is the rest-mass and internal energy that are equivalent.
All good and well, but then why does kinetic energy not change the mass? We could say that it does by defining a relativistic mass, but again, avoid the temptation! That would be a definition of mass that depends on frame of reference. Instead, notice that the mass measured in the object's own frame never depends on how fast it is moving. This makes it easy to define, easy to measure, and everyone everywhere can quickly determine the value and agree on it. It captures the properties of the object itself, and not situational circumstances like how fast the observer is moving which do not affect those properties!
Want to determine the mass of something moving fast? Easy! Let it hit something and come to a stop. Then its kinetic energy is reduced to zero in your frame, and must have been converted to something else. The total energy reduced from [math] to [math]. The difference is [math]. That's how kinetic energy is defined in relativity! (The Newtonian formula, [math], is wrong at high speeds.)
Notice the mass did not change when we stopped it. It was m before, and it is the same m after.
An outstanding question at this point might be why we speak of the masses of particles increasing in particle accelerators. By observation, the products of colliding two relativistic particles can have greater mass than the components. Doesn't that mean the high velocity of the components increased their masses? The answer is no! When they collide (let's say perfectly inelastically), their kinetic energy drops to zero. This converts it, very briefly, to internal energy. That internal energy is then what provides the mass for the resultant collision products by E=mc2.