Is the 2"x4" backing strong enough to hold the server rack
You're asking about the 2x4. The issue is the fasteners. You won't break the lumber. 

You didn't say whether you piloted to the lag screws, but assuming that you did not, or that you did so properly, they should hold well. That said, I'd have probably either used a 2x6 or a small sheet of 3/4" plywood. My reasoning is that a 2x4 is narrow enough and thick enough that it tends to roll. There's a lot of leverage (torque, really) being generated. A taller and/or thinner backer board would have less leverage affecting it. 

The same is true of the screws used to mount the server rack bracket. Pilot properly (slightly smaller than the screw shaft diameter--not the thread diameter) to be sure they don't pull out. Pine is soft. 

Also make sure that everything pulls snugly together. Gaps lead to movement, which amplifies pullout forces by transitioning them from shear to tension.

Finally, your carpentry looks good, but a hex screw in wood should almost always have a washer behind it. This prevents damage when tightening and increases load distribution. If you have those lags tight enough they'll pull into the wood somewhat.