Information has more than one sense. The SEP article on information provides half a dozen.
The Shannon concept of information, which you refer to, is rooted in communication theory. It is naturally interpretable as the maximum amount of 'content' that can be conveyed in a message. More generally, we can think of it as a way of quantifying the concept of distinguishability. If I can distinguish between a switch being in the up position or the down position, then I possess information. To be precise, I possess one bit of information. Likewise, if I can distinguish between here and there, off and on, this and that, etc., then these are all examples of information that I possess. Information can be correlated, so it is not simply additive. If I can see by looking at the light switch whether it is up or down, and also see by looking at the light whether it is on or off, this does not sum to two bits of information, because the two are strongly correlated. Not perfectly correlated, because the light may be off due to a power failure or a circuit breaker tripping out, or it may be on because a mischievous person has shorted the switch out with a nail. But Shannon information theory, coupled with Bayesian probability, can allow us to express relationhips in such a way as to allow us to quantify how much distinguishability I have. Although I speak here in the first person, information in this sense does not have to be possessed by a human agent. One could speak of a measuring instrument or a computer having information. Information can be understood as a quantitative measure of the number of distinctions represented by a set, or a distribution, of possibilities. I believe this is the sense in which the concept of information is most commonly deployed in physics and biology.
According to some theorists, it is no accident that Boltzmann entropy and Shannon entropy have a striking parallel, as you put it. It is possible to interpret statistical mechanics as a purely statistical theory based only on some simple properties of matter at the microscopic level. Arieh Ben-Naim has written several books on entropy from this perspective. In particular, in his "A Farewell to Entropy" he shows how the Sackur-Tetrode equation for the absolute entropy of an ideal gas can be derived from information theoretic considerations. On this view, the second law of thermodynamics is not so much physics as an application of Bayesian statistics. This view is not undisputed, however, and more detailed consideration would take us into ergodic theory.
At the level of human cognition, we think of information as relating to semantic concepts such as meaning and truth. A message may contain information in the Shannon sense, but if it is the ciphertext of an encrypted message and I don't have the decryption key, it conveys no meaning to me. Also, a message may simply be false: Floridi (The Philosophy of Information) prefers to restrict the term information to things that are meaningful and true.
I think you are correct in saying that there is no generally accepted way of combining these two concepts of information. Dretske (Knowledge and the Flow of Information) made a valiant attempt at it, but it only goes so far. One might say that the problem of relating information in the purely mathematical sense to information in the semantic sense is an example of the problem of relating extension to intension. If we had a systematic way of doing that, logic would be a lot simpler and more powerful.
Speaking of logic, information theory suggests a way of interpreting certain common concepts within logic. A valid argument might be thought of as one in which all of the information in the conclusion is present in the premises. A tautology is a sentence that conveys no information, etc. Information here might be understood in terms of the set of logical possibilities that are excluded. If you are interested in this, a couple of useful papers are David Ellerman "An Introduction to Logical Entropy and its Relation to Shannon Entropy" International Journal of Semantic Computing, (2013) 7(2): 121-145; and Jon Barwise "Information and Impossibilities" Notre Dame Journal of Formal Logic (1997) 38(4): 488–515.