If you correctly understood what people told you, then it is not clear that they understood the matter any better than you did at that time. As quoted, they certainly did not make it very clear. However, to begin with you must understand that part of what you are enquiring about cannot be clearly understood except in mathematical terms. For a start, it is very difficult even to understand what "energy" is in terms of verbal descriptions. Of course, the same applies to mathematical descriptions, but at least "correct" and applicable mathematical descriptions have "correct" implications, meaning that one can reliably deduce the logical consequences of events. For instance, one can deduce that something like a gram of the mass equivalent of the Hiroshima bomb was released during its detonation.

You said that "Mass is a characteristic of matter”, but that is not really definitive. Apart from various relativistic considerations, if you were to measure the mass of a complex structure, such as an atom, and compare it with the mass of its individual components, you would find a discrepancy. Whether the discrepancy is positive or negative would depend on such things as the size of the structure. For example, a plutonium atom would have a greater apparent mass or “mass equivalent” than the fragments after it had split. Conversely a helium atom would have a smaller mass equivalent than say, two deuterium atoms in the same inertial frame.

In dealing with such distinctions it often is less confusing to speak of "mass equivalent". It also may be useful in context to refer specifically to "rest mass". The problem is that most systems comprise both matter and energy, both of which contribute to their respective "mass equivalents".

First of all, in these terms one could regard E =mcc as simplistic, where m stands for rest mass. For example, what do you do about light, which has zero rest mass? A better statement would take the form of something like: EE=Pcc+mmcccc, (or E squared equals momentum (P) times c squared plus m squared times c to the fourth).

Under everyday conditions, you can see that the momentum at everyday speeds will be negligible for most purposes, compared to the other terms in the equation. Conversely, in a system without matter, and correspondingly with zero rest mass, such as a photon, the energy amounts to the square root of its momentum times the speed of light.

So for the first point, energy does not have mass except in a simplistic sense; it is better to speak of its mass equivalent.

Although the two correspond in many respects, it is misleading to speak of matter and mass equivalent, or even mass, as being identical to energy. Quantitatively, this may be true mathematically in suitable senses, but in most practical terms there are complications. Energy occurs in too many forms, many of which affect many everyday considerations. By way of analogy, you might say that a human is so many kilograms of each of: hydrogen, carbon, nitrogen, oxygen, phosphorus, calcium, etc. Now, if you measured my bodily contents in those terms, that certainly is what you would find. It does not imply however, that if you took those components in a container and inspected them you would find a human present, let alone me. Analogously if you measured the mass of a a proton it would be equivalent to just about a giga electron volt, but if you somehow gathered a giga electron volt in the form of photons in as much space as occupied by a proton, that would not give you a proton, nor the equivalent in electrons.

As soon as you deal with everyday matter such as protons, electrons, neutrinos, atomic nuclei and the like, you find yourself dealing with things like hadron number, lepton number, electric charge, spin, and so on. These things are conserved quite apart from the conservation of matter or energy in general, so that whereas one could argue about the equivalence of mass and energy, you flirt with serious confusion as soon as you speak of the equivalence of matter and energy.

One way of looking at mass, or for that matter, at mass equivalent, is in terms of its gravitational effects. The gram of energy released by the bomb I mentioned, would have exactly the same gravitational effect as any other gram. However, if you consider the difference between its gravitational effect and its effects in terms of a less notorious equation than E=mcc: F=ma, you will deduce that we are dealing with altogether different orders of magnitude. Specifically we derive the mass equivalent as simply m=E/cc. There is nothing surprising about that derivation, but it is not one that we have to contemplate very often. As you could deduce from its magnitude alone, it is indeed, as you put it, a little tricky to measure.

Be careful though in speaking of "anomalies" in this matter. The mass equivalence of matter and energy are extremely, even spookily, consistent. It is not so much that they "explain" dark matter and the like; they do not. It is just that any theory or prediction that violates their implications is deeply suspect and generally would be dismissed out of hand unless there were special reasons for giving it serious attention.

As you can see, your original views were not in general unreasonable, but the matter is not simple. There is a pretty good and fairly simple discussion of many of the points, for example in Wikipedia at:

http://en.wikipedia.org/wiki/Conservation_of_energy

and:

http://en.wikipedia.org/wiki/Mass-energy_equivalence

Have fun,

Jon