Dear Jonathan,

I don't know my explanations were enough to solve your questions.
If I were you, I would try to not be so focused on the math beyond the Duhamel integral. It is just a tool to calculate what we need.

The DLF vs. Frequency plot is rather an indicator (of dynamics) sensitive to the frequency content of the applied load, much more useful in practice than a plot of "exact" DLF vs time, for example.

In case you downloaded the book I've mentioned or just seen Coade article, you may see that in the simple case of the "perturbation" as a force which is suddenly applied and remains constant indefinitely, DLF has expression 1-cosωt.

However, in practice we are not interested to know the dynamic response vs time (or DLF vs time) but to see only the maximum value of the DLF which is really of interest.
In the case considered, this maximum is 2. You may say that for each ω the DLF is 2 (because mathematically I can find the time when cosωt=-1 for each ω considered but more important is that in this case DLF=1-(-1)=2 and this is the maximum possible since -1<=cos<=1) and this is told as "always the DLF is maximum 2" - for the case when a single instantaneously "hit" is applied ,I would add (my last "addendum" is too often forgotten!). Moreover, as you can see, one can realize that this "2" is not really related to a particular system but to any system with single DOF which is subject to this kind of "perturbation".

I hope this example gives you DLF spectrum feeling you needed or at least why is a "spectrum". If not, I guess Mr. Diehl or some other contributor can explain much better than me.