What do physicists mean by "information"?

What do physicists mean by information?

Every now and then I'll read a book by a scientist trying to explain a field or subfield or subproblem to laypeople like me (by Randall, Susskind, Greene, to name a few of the recent). I almost always like these books, but Sean Carroll's From Eternity To Here is the first that's really clicked for me. I wouldn't remotely claim to understand it. But I got enough while reading to follow what it was doing as it went circling and chasing a few basic questions. Fundamentally: our universe, or our part of it, seems to start with low entropy, the entropy increasing over time. (Other possibilities are explored.) Without this low-entropy start, time wouldn't have the direction it has. Why did we start with low entropy?

This actually encompasses a whole bunch of questions and contentions. I read the book several months ago, in spare time, needing all three library renewals and finishing on the last day, not thinking concentratedly enough to master one part before moving onto the next. I couldn't have mastered the parts anyway. At one point I copied down the following passage, writing underneath, "If I want to test whether I've understood the book so far, I could look at this passage and ask myself to explain it: to fill in the background, to describe how his argument got here. I don't think I can right now." I understand it even less a few months removed:

In that sense, the irreversibility that crops up when wave functions collapse appears to be directly analogous to the irreversibility of ordinary thermodynamics. The underlying laws are perfectly reversible, but in the messy real world we throw away a lot of information, and as a result we find apparently irreversible behavior on macroscopic scales. When we observe our cat's location, and our own state becomes entangled with hers, in order to reverse the process we would need to know the precise state of the outside world with which we are also entangled, but we've thrown that information away. It's exactly analogous to what happens when a spoonful of milk mixes into a cup of coffee; in principle we could reverse the process if we had kept track of the position and momentum of every single molecule in the mixture, but in practice we keep track of only the macroscopic variables, so irreversibility is lost.
--Sean Carroll, From Eternity To Here, p. 255

My question about information is this: We use the word "information" for a whole lot of fairly different stuff, but I think of it as including, e.g., "The test tomorrow is at 1:00 PM." But when a physicist like Leonard Susskind is talking about "conservation of information," I don't think he means that something like "The test tomorrow is at 1:00 PM" is preserved — e.g., would survive planets being absorbed into black holes and the black holes dissipating. But when he uses the phrase "conservation of information" he does indeed mean that all information is conserved after planets are absorbed and black holes dissipate; so I'm surmising that the sort(s) of information he's talking about differ from the type of info that would include "The test tomorrow is at 1:00 PM." But maybe the latter is included.* If so, how would that work?

*I read The Black Hole War and of course have forgotten most of what I read.