I think there are pedagogical advantages to discussing both the raw numbers and the absolute values and I think the benefits of both explain why they both show up (sometimes in the same text, even).
Each elasticity number gives two bits of information. First, the absolute value with respect to 1 and second, the sign. Now, clearly, if you had a negative elasticity, you could compare it to -1. However, it becomes somewhat difficult to teach when using phrases like "greater than" or "less than" -1 to discuss a good being (in)elastic, since "greater than -1" is actually inelastic if the elasticity is negative. It is much more intuitive to be able to discuss the ratios of percent changes if "greater than" does in fact mean that the top is bigger than the bottom and vice versa for "less than".
Of course, there is also a bunch of information tied up in the sign of the elasticity. We get the Law of Demand out of own-price elasticity, we get compliments/substitutes from cross-price elasticity, etc. So it is important to still make sure students understand the importance of the sign.
When I am teaching, I try to discuss both parts explicitly, but make clear that the elasticity itself includes the appropriate sign. I think most books are trying to capture these two bits of information in one way or another. In any case, the formal definition of elasticity should include the sign, but if one is just talking about how elastic a good is, the absolute value could be reported (with the note that it is the absolute value of the elasticity, not the elasticity itself).
As for MRS, it's usually not the absolute value, per se, that we report, but rather the negative of the derivative dy/dx. This is quite standard, since it has the intuitive interpretation of the consumer being willing to give up so many units of x for so many units of y. Since indifference curves are usually convex, this derivative is negative, thus changing the interpretation (and intuition) somewhat if we don't negate it.