Good question, and good point. I read a definition of information a while ago (actually it was Judea Pearl's), which defined it in the usual log-negative way, but in terms of the probability of states that you care to distinguish.

I mean it's obviously sensible. In the context of binary information in a computer, for instance, we (generally) don't care about exact voltages, we group it all together as 0 or 1. And talking about genetic information, we don't (usually) care about, say, if a slightly unusual isotope has snuck in in one of the usual nucleic acids, since that "information" isn't copied in the usual way.

It's very obvious, but it's also more profound than it looks. Even for something as seemingly purely mathematical as information, there's actually teleology, assumptions about what matters, baked in every time we apply it.

This is a great question, but I'd answer that paying _some_ attention is better than paying _no_ attention at all. Some people study and train for years and reportedly they achieve full happiness, but certainly that is not my case and, to be honest, it's hard. ymmv.