How then can you assert that new information cannot be added by random mutations if you're not even sure what new information is?
Yes, of course I do. But this is based on my understanding of information.
My understanding of what new information constitutes does not place any requirements on the origins of said information. To me, new information amounts to a new function. New information does not refer to the net total of information, rather it is a descriptor of information that had not previously existed. In other words, new information is simply functional information that was not previously present. It may or may not involve the loss of old information. For example, a population of cells has a functional set of proteins X. If it can be demonstrated that the functional set X did not include function Y, then the appearance of function Y indicates new information. New information must have a function, or its definition serves no useful purpose. This is my opinion. I'm sure if I sat down and wrote for a longer period I could come up with a more concise definition, but this is just something to work with.
The question then arises as to how to measure the summation of information in a system. I am not an information theorist by any means, so if anyone is please correct me. It would appear to me that information content can be measured by the functional outcome of said information. This functional content of a gene is not necessary related to genomic length. For example, a gene which codes for a small, simple protein with function X carries just as much information as a gene which codes for a large complex protein that performs function X. If you and I are having a text conversation about our heights, you may say, "I am 5 feet and 11 inches tall." I may convey the same message as "5-11". Your string contains more binary than mine, but they both have the same information content. Thus, genomic length is not a good indicator of information. I could go on, and I may pick this up again this weekend, but I have a renal physiology test tomorrow which is going to bury me.