This may be a stupid question, but what do people usually mean when they refer to a mathematical text as being "rigorous"? Does it mean that everything is strictly proof-based rather than application-oriented?

Generally that's what it means. And also when proofs are presented, a rigorous book will go through it fully, whereas a less rigorous one might just sketch out the main ideas of the proof and leave out some of the nitty gritty details (i.e. it's less rigorous to talk about "continuity" as "you can draw it without lifting the pen" as compared to the epsilon-delta definition, but epsilon-delta is pretty detailed and for intro calculus for non-mathematicians you don't really need it).