I started in cell and molecular biology, but moved to applied math and computer science, so perhaps I can give my thoughts.
Personally, I found that biology learning was more about big ideas and visualization. The first is because biology has an extraordinary amount of details (e.g. Molecular signaling pathways, biochemical properties of different molecules, the location of different foramen in certain fossils, etc...), so it is truly impossible to remember them all. You should focus on learning what to look up, not what is actually occurring. Something you work on will be gradually memorized on its own, so don't spend effort memorizing things you can look up (you're not an undergrad :) ). So don't remember that histone acetyltransferases use acetyl-coa to act upon the NH3+ groups of lysines. Just remember that those enzymes modify chromatin in a way that alters transcription, thus changing the cell and its behavior. The second (visualization) is because it is very hard to understand biology without pictures! So use them, not text (no kobayashi and nomizu style). Mathematicians like symbols but (at least in the molecular case) there are too many of them (and they are mostly useless names... e.g. P53). Much of biology is about seeing visual results (blots, fossils, medical images,...) and interpreting them.
Note that (I think) the first suggestion is harder in biology than in math, because math partitions naturally into theorems, corollaries, lemmas, and their applications (roughly in order of importance of what you need to know :) ). Biology as a field is more of a fast moving chaos of observations and changing inferences. But of course there is a constantly growing core of the field that is not in flux.
One good thing about biology is that one can be an expert in one area but know little about most others. Math is a little more intertwined. So there's no need to read a "general bio" book. Just read an intro book to your target topic (e.g. Ecology, evolution, cell bio, molecular biophysics, bioinformatics ...). They will cover the relevant parts of other fields, usually. I know genetics experts who know basically nothing about cell trafficking for example. But you will definitely need some background to read the latest literature (i.e. Don't skip straight to Cell papers).
One issue that occurs throughout science, but also in math, is that learning is a process of improving models. In math, we might start with linear algebra, then basic manifolds, and eventually with, say, finsler spaces. In physics, we go from Newton to Einstein. In biology, we start with a basic model and set of rules, but gradually add exceptions and details to it. I think this is where much confusion can arise, because things that were correct in the simplified model might be wrong in the better, more complex one.
Also, models are being updated at an insane rate in biology. Often that means that previous models were wrong (though the evidence underpinning them is not); hence definitions and results might change over time! Science improves over time, and this is a good thing. Of course, in math, once something has been proved, it will never change. (Although definitions, or at least notation, in math can be very inconsistent too). Of course, if there are inconsistent definitions, you can ask here and see what people think. I will agree that biologists are less concerned about detailed rigor than being correct overall. Generally scientists will either define such things themselves or cite a reference with their chosen definition.
My suggestion for your question is to learn the basic widely accepted definitions first, then go to the literature. Often it is the case that different groups think different things. Most likely, one or both will be proven wrong eventually. Each paper is likely to be internally consistent; the only ambiguity is usually in areas where no one knows the correct answer or interpretation. Two groups in conflict usually have reasonable hypotheses consistent with the evidence, and if they can't decide how to mesh their ideas, probably we can't either. So don't worry about it too much.
So when a topic has multiple interpretations that are possibly true, use your math background and consider it from a Bayesian perspective :) most likely all the interpretations are at least partly true, and have some non zero probability of being true. And as you read more papers, update your posterior. :)
Also, don't worry about talking to biologists (as the other commenter states). In fact, among the sciences and engineering fields, my opinion is that biologists are on average the friendliest and least arrogant :)