What about a photon? If I use light to transmit information have I given it mass?
Photon's stress–energy tensor causes similar effect on gravitational field as tiny amount of mass equivalent to the energy of a photon.
Photon has momentum, zero rest mass. Energy mass equivalence says that you can annihilate mass to create a massless particle. This is true for all the observers.
Relativistic mass is a perceived mass dependent on the observer.