No, they have energy. Relativistic mass is a pretty damn confusing way to represent momentum. You wouldn't say that the mass of the electron (~511 KeV/c^2) is variable just as you wouldn't say the mass of the photon (~0 in whatever units you want) is variable.
So yes, they do have relativistic mass, and you disagree with the use of that term. According to your link, about 60% of modern authors avoid using it. Maybe it's getting phased out, maybe it isn't - it's controversial. There's no scientific consensus, so it's not fair to get on anyone's back about using it.
It seems to be a semantic argument. Mass is a word we define, and "relativistic mass" equations match observed reality. Even with photons. It works in terms people understand - things with momentum have mass. Momentum is velocity times (relativistic) mass.
Or, we can describe the same thing using rest mass, instead of calculating the relativistic mass separately:
(Sorry Darkling, click to view)
Note that the second equation doesn't actually help with photons, since they (probably, unobservably) have 0 rest mass. 0/0.
I just don't see the need to vehemently insist that photons never have any mass, when "relativistic mass" accurately and conveniently describes how we observe them behave. It's semantic.
Quantum mechanics introduces the idea that light can be viewed as a collection of "particles": photons. Even though these photons cannot be brought to rest, and so the idea of rest mass doesn't really apply to them, we can certainly bring these "particles" of light into the fold of equation (1) by just considering them to have no rest mass. That way, equation (1) gives the correct expression for light, E = pc, and no harm has been done. Equation (1) is now able to be applied to particles of matter and "particles" of light. It can now be used as a fully general equation, and that makes it very useful.