Is massenergy an outdated concept? Do physicists working on relativity still use the concept of massenergy? Do they still say that mass and energy are the same thing? Einstein coined the term "massenergy" to make clear, he said, that mass and energy were the same thing, by using the one word for both.
But he also said that mass increases with speed, and I have seen many times claims from credible people saying that that is an outdated concept, and the modern way is to say that mass does not increase with speed but say that momentum does increase with speed, and say that total energy does increase with speed.
I've even seen credible people writing that $E = mc^2$  is not true.
 A: Mass and energy are still equivalent. Nothing has changed on that front since 1905.
What has changed somewhat is the English language as used by physicists. Since the separate words "mass" and "energy" exist, it's convenient to use them for different things. It's a common convention these days to use "mass" for rest mass-energy, and "energy" for other sorts of mass-energy. Of course if you write these $m$ and $E$ then $E\ne mc^2$, though $E$ may equal $\sqrt{(mc^2)^2+(pc)^2}$ or $\sqrt{(mc^2)^2+(pc)^2}-mc^2$.
At the most fundamental level, there is no clear distinction between rest mass-energy and other mass-energy. Much of the rest mass of a proton is kinetic energy, and even the rest masses of fundamental particles don't look very fundamental in the Standard Model. "Rest mass" is just a way of saying that we don't understand or don't care at the moment about an object's internal structure, but it's still a useful concept.
(I rejected a proposed edit to this answer that replaced "mass-energy" by "massenergy" because no one uses the term "massenergy" unhyphenated, as far as I know.)
A: Maybe answering too late, but I had read the question when you posted and I was taking my time to think about it.
I am not a physicist just a physics enthusiast. And I am huge fan of Einstein, I believe Einstein thought something different from us.
Your question was "Is massenergy an outdated concept?"
Mass-energy isn't outdated concept. . "People" (Relativist) currently use that concept in their daily life. What is outdated that is relativistic mass. To understand why it is considered outdated one must know what physicists mean by "outdated".

As such, I would consider 'relativistic mass' to be outdated because it is a mathematically valid term that was used more often in the past and that it is unessecary to learn to obtain a complete understanding of present day relativity. ~ Physics Codidact

So relativistic mass is mathematically correct but not helpful in your daily life.

Brian Greene once said, "you can prove lot of things mathematically but not all of them will be helpful in your daily life" (he used "correct" rather than "helpful" and it's not a quote I just listened it in one of his lecture)

Physics is more about physical things. And you can get to the edge without using the "relativistic mass" context. Einstein didn't talk about Relativistic mass in 1905's paper. He latter used the term over and over again.

A hotter body looks more massive than a colder body. (With relativistic mass in mind) since, particles in hotter bodies moving/vibrating faster than a colder body's particle that's why according to $m_{rel}=\gamma m_0$, one can say that the mass is looking greater for the relativistic mass.

I suspect one can express the behavior some other way, (with no knowledge of GR, QM)  I will say that the behavior may be expressed well from QM (I will edit the answer whenever I am there) (I would request a professional to say something about it in comments :) ).
I will just quote some important context from "the book".


In his review article on special relativity from 1907, Einstein shows that a body of mass μ that has absorbed an amount of energy Eo as measured in its rest frame executes motion, in an inertial frame relative to which it moves with some velocity, as if its mass M was given by the expression (Einstein 1907b, p. 286):In a footnote, Einstein explains the convention, which he had already adopted in an earlier paper (Einstein 1907a, p. 250), of using “the subscript ‘o’ to indicate that the quantity in question refers to a reference system that is at rest relative to the physical system considered” (Einstein 1907b, p. 286).After 1907, Einstein’s notation crystalizes so that by 1921, in his Princeton Lectures, Einstein expresses his famous result by writing (Einstein 1922, p. 46):


A very small part of this energy resides in the thermal motions of the molecules constituting the particle, and can be given up as heat; a part resides in the intermolecular and interatomic cohesion forces, and some of that can be given up in chemical explosions; another part may reside in excited atoms and escape in the form of radiation; much more resides in nuclear bonds and can also sometimes be set free, as in the atomic bomb. But by far the largest part of the energy (about 99 per cent) resides simply in the mass of the ultimate particles and cannot be further explained. Nevertheless, it too can be liberated under suitable conditions, e.g., when matter and antimatter annihilate each other (Rindler 1991, p.75).


When viewed from a Newtonian perspective, and assuming that the vessel containing the gas is itself massless, the mass of the vessel of gas is simply equal to the sum of the masses of the molecules. From a relativistic point of view, this last assertion states incorrectly that the rest-mass of the vessel of gas is equal to the sum of the rest-masses of the molecules. Yet, from a relativistic point of view, the rest-mass of the vessel of gas is equal to the sum of the rest-masses of the molecules plus the kinetic energy of the molecules divided by c2. Since, according to the Kinetic Theory of Gases, the temperature of the gas is proportional to the average kinetic energy of its molecules, if the gas temperature increases or decreases, the rest-mass of the vessel of gas increases or decreases accordingly by a tiny amount.


Einstein’s original derivation of mass-energy equivalence is the best known in this group. Einstein begins with the following thought-experiment: a body at rest (in some inertial frame) emits two pulses of light of equal energy in opposite directions. Einstein then analyzes this “act of emission” from another inertial frame, which is in a state of uniform motion relative to the first. In this analysis, Einstein uses Maxwell’s theory of electromagnetism to calculate the physical properties of the light pulses (such as their intensity) in the second inertial frame. By comparing the two descriptions of the “act of emission”, Einstein arrives at his celebrated result: “the mass of a body is a measure of its energy-content; if the energy changes by L, the mass changes in the same sense by L/9×1020, the energy being measured in ergs, and the mass in grammes” (1905b, p. 71). A similar derivation using the same thought experiment but appealing to the Doppler effect was given by Langevin (1913) (see the discussion of the inertia of energy in Fox (1965, p. 8)).

(there was some more but they will make the answer bigger)


It is not good to introduce the concept of the mass $m_{rel}= m_0 \gamma$ of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion.


"The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."

Further reading :
Do photons have any relativistic mass?
Why is there a controversy on whether mass increases with speed?
A: When you add energy to a system, the mass of the system increases. Einstein suggests then when calculating the curvature of space-time energy should be included as a causative factor. This would suggest that there is mass associated with energy.
A: "Mass" and "energy" are two different things.  $M^2=E^2-P^2$ is the invariant length squared of the four-vector $(E,P)$, and energy, $E$, is it's zeroth component.
In the rest system, where $P=0$, they are equal in magnitude.
It's like the distance, D, to the top of a tree and its height, H. They are different but are equal in magnitude if you stand at the base of the tree.
There has been much confusing language about this, even by Einstein, but I think he eventually corrected himself.
