You are right in principle but off in magnitude. When I referred to cycles I did not mean frequency or pitch (a vibration rate), but rather to the total number of cycles of vibration without regard to how long that took. That's what I was interested in, and the Wiki Commons plots you borrowed show how you work this out.
In electronics engineering we usually use Q in resonant circuits to predict this. Q is the amount of energy at the start of the cycle divided by the portion of that energy that is lost by end of a cycle. Your equations use a different measure of the loss factor called zeta (ζ). It is the ratio of the loss that you actually have per cycle to the loss per cycle that produces critical damping, the minimum level of loss that prevents ringing.
What I did was replicate the Wiki Commons plot in Excel so I could change zeta to different values. The first plot is the lower ζ=0.3 plot which is the lower plot in Wiki Commons illustration, but with the time scale widened. The next is ζ=0.1, same as the upper Wiki Commons plot. I then went on to give you ζ=0.03 and ζ=0.01 so you can see that as the loss factor gets smaller the ringing cycles take longer and longer to decay.
Take a listen to the tuning forks in this You Tube video
. You can hear they sustain for a number of seconds before they lose much volume. From that example, if I suppose a tuning fork tuned to A above middle C (440 cps) takes 5 seconds to lose half its magnitude, then it has a zeta value of about ζ=0.00005. My last two plots show what this looks like on both the same scale as the first plots and then on a much wider scale of cycles.
As we both mentioned previously, the barrel steel is softer and won't ring as long, but keep in mind that a brass bell is softer than barrel steel and it still rings pretty well. Maybe only a fifth as long as the tuning forks. My expectation is the barrel steel won't be any worse than that, though I'd have to check on it. The bottom line is I expect loss in magnitude to be only a few percent by the time the bullet exits, and not major losses in magnitude.