Does light have gravity or Gravitomagnetism? We all agree that light has no mass yet it is affected by gravity.  According to accepted theories I have seen light itself is also said to bend space meaning that it causes gravitation.  This would seem to contradict the very definition of gravity as an attractive force between two masses.  It also seems to cause contradiction as the energy of light varies depending on the observer.
I have heard of a phenomena called Gravitomagnetism which is said to be an analog to magnetism.  As I understand it, this is a force which pull masses in the direction of other masses that pass by similar to how magnetism pull charges in the direction (or opposite direction) of electrical current.
So my question is this - does light in fact have gravity or does it just have Gravitomagnetism meaning the ability to pull nearby object in its direction of travel rather than toward the light-beam.
 A: 
So my question is this - does light in fact have gravity or does it just have Gravitomagnetism meaning the ability to pull nearby object in its direction of travel rather than toward the light-beam.

Here is a list of the abilities that light has, and after each ability in parenthesis there is the name of the ability.

*

*Light has the ability to pull a nearby massive object towards itself. (attractive gravity)


*Light has the ability to pull a nearby massive object in its direction of travel. (gravitomagnetic induction a.k.a frame dragging)


*Light has the ability to not pull nearby parallel light towards itself. (attractive gravity and repulsive gravitomagnetism together)


*Light has the ability to pull nearby anti-parallel light towards itself. (attractive gravity and attractive gravitomagnetism together)


*Finally, I think two passing anti-parallel light pulses should induce some gravito-motive forces on each other, but after they have passed they should be like before the passing, except for some small change of direction. Like two passing electrons induce an electro-motive force on each other.
A: “Gravitomagnetism” is just one formalism for understanding General Relativity. In GR, the density and flow of energy and momentum (expressed as the energy-momentum-stress tensor) is what causes spacetime curvature and affects the motion of particles. Whether you are talking about massive or massless particles, their motion helps determine the gravity they cause.
If you want to see how the velocities and accelerations of two massive particles affects their mutual gravitational accelerations, look at the Einstein-Infeld-Hoffmann equations. I have not seen analogous equations for massless particles, but they probably exist.
A: The answer by G.Smith is to the point, here  I stress the separation between the General Relativity mathematical model and the Newtonian mathematical model of gravity.
Physics is a scientific discipline where observations and measurements are fitted with mathematical models which describe existing data and successfully predict new values for new boundary conditions. When this happens one says that the model has been validated.
If new experiments and observations should falsify the model, one will have to re-examine the assumptions and even search for a new model.
In the particular case of gravity, Newtonian gravity developed from fitting mathematically observations and measurements up to  two centuries ago, using classical  mechanics , and a single axiom/law , Newtons law. 
To first order the assumption that gravity is always attractive fits the data for velocities much smaller than the velocity of light, and masses much smaller than the earth. This is true to first order in the mathematical calculations, because, by a brilliant extension of the mathematics of electromagnetism, Einstein formed a mathematical theory called General Relativity, which gave predictions that were validated by better measurements and better observations. This means that even for using the GPS system accurately on earth, one needs general relativity corrections to the first order orbits of the Newtonian model.
It has to be stressed that a new physics theory describing the same data, must be consistent at the limit of overlap with the old theory, because the old theories are a mathematical mapping of a great number of data, and are a type of a boundary condition to any  new  theories, there has to be consistency in the overlap regions of phase space, and the definitions of variables.
You say 

This would seem to contradict the very definition of gravity as an attractive force between two masses. 

Gravity has to be attractive only at low energies and masses, because that is what has been measured and observed.
Within the above context, the question

So my question is this - does light in fact have gravity or does it just have Gravitomagnetism meaning the ability to pull nearby object in its direction of travel rather than toward the light-beam.

is confusing the region of validity in the behavior of  light/photons, between two theories, the classical and the GR.
In classical theory, even with quantum mechanics, photons have no mass, so have no Newtonian attraction.
In GR masses are the manifestation of the  tensor defining the energy and momentum distributions of the system under study. Thus for the mathematics of GR as light has energy , it contributes to the tensor and thus distorts space, and this has been observed in astrophysics.
Force has no meaning at the energies and masses of GR. It is a consequence of the space distortions that can be derived from the mathematics of GR. 
Gravetomagnetism 

Gravitoelectromagnetism, abbreviated GEM, refers to a set of formal analogies between the equations for electromagnetism and relativistic gravitation; specifically: between Maxwell's field equations and an approximation, valid under certain conditions, to the Einstein field equations for general relativity. Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge.

your suggestion:

just have Gravitomagnetism 

posits a new model, and in addition to  not being a mainstream model, it would need a lot more theory to fit  high mass high energy data  the way GR does.
