["Live Rock" is one of those terms that has come to mean different things to different people. More often now, it used as a marketing term.
Back when the Berlin methodology first came about(and if you are using "LR" and a skimmer only, you are using the Berlin method), "Live Rock" meant any rock which had "live" resident bacterial populations.
Through competitve marketing initiatives from various retailers, the term has sorta evolved to mean, full of corraline, macros, corals attached, and all kinds of nifty things. But as far as the Berlin method is concerned, it refers to resident bacterial populations only.
Obviously the more porous the rock, the larger the bacterial populations that can reside there. This rock, for those using the Berlin Method, is the most desirable, and the most expensive.
To help defray some of the expense in setting up a system, often times people will build the lower structure of their system using "base rock". Base rock, is nothing more than a ambiguous subjective term, but in this context, it usually means, nonporous(or less porous than the other rock you are using...). It's less desirable, and less expensive. The solid limestone slabs you see on the shelf at your LFS is a perfect example of this. Not very porous, and with no resident living bacterial populations. But it provides a cheap foundation to begin the building of your reef structure.
Where rock is found in the reef has nothing to do with it being considered "live" or "dead". In fact, I'd submit that even the most dense, least porous rock you could find in the ocean, would still be considered "live" when harvested, and if I took a chunk of prime fiji out of my tank, and let it sit for a year in the sun, it would be considered "dead".
The term "live rock" is often one where the hobbyist gets lost between methodology and marketing. Aragocrete while "dead" sitting in your garage after mixing it up, will likely be very much "alive" soon after adding it to you tank, meaning, bacterial populations will move in and set up shop.]
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