Riyas and Richard, many thanks for your replies. Re: Richard, indeed, what I meant to say was "a line that crosses a buffer's boundary is not useful/important."
Therefore Riyas's solution should work. Unfortunately I receive an error message that "arcgis has encountered a serious application error" when I try the spatial join (lines = join features; buffers = target features, join operation = one-to-many, match option = completely contains). I've tried switching 'background processing' (Geoprocessing > Geoprocessing options) on and off, but that doesn't help, I always receive an error message and ArcMap crashes.
It is worth mentioning that he visual examples I've provided above are small pieces of the larger dataset I'm dealing with. I don't think the size of the data I'm working with should cause the computer to crash: in all, there are 599 buffers and 2,895 lines. I'm a bit confused about the error message and plan to call ESRI tech support to see if they can be of any help.
Since the spatial join does not work (at least for the moment, until I can resolve the error message issue) I've gone back to the Modelbuilder exercise at the very top. I got it to work, although I must say it is a rather inefficient method to get the information I want. To the output of the Summary Statics GP tool I added an 'add field' tool and named it BufferID, then I added a 'calculate field' and drew an arrow from the light blue 'Value' oval and made this the expression. Then I added the 'append' tool where the target dataset was a Table I had to create separately.
For the 599 buffers and 2,895 lines, this model took 34 minutes 45 seconds. In other words, about 3.48 seconds on average per iterated feature (3.48 seconds * 599 iterated features = 34.75 minutes). If I could do a spatial join, I am sure the process would run in under 2 minutes! Frustrating...
I may try Richard's solution, which I find quite inventive, to cut down on the processing time.
Thanks again!