I agree there's the obvious inner voice that meditation can definitely come close to eliminating, even outside sitting. It's the unbidden inner voice we experience like crazy when we first start meditating. Monkey Mind, if you will. The power of meditation is making us more aware of it when it first begins to arise so as to give us the power of veto and not allow it to take us places that aren't productive.
Then there are more subtle functions of mind that aren't really an inner voice but can certainly have a similar effect and be directly experienced. Intuitions, desires, aversions, feelings and the like. All non-verbal.
Then there's a very deep and more subtle aspect of mind also at work that I'd associate with the Freudian ego. It is part of the subconscious and it really has to be else all its processing requirements would probably take up all of our conscious processing power. In my way of thinking it's what gives rise to the non-verbal experiences above and probable the verbal inner voice too.
That last aspect of mind was what the gentleman was referring to when he told me to be ever vigilant. It will latch onto things that might start out as virtuous or altruistic and bend them to its own agenda. As an example he said just Google Buddhist monks and sexual scandals, especially where there's a power relationship such as teacher/student, guru/follower, etc.. It goes to their heads, so to speak, and I'll leave it to you to decide which one.
I always use to joke about it being okay to talk to yourself, and even okay to answer back, having a conversation. When you start interrupting the conversation is when you should start to worry. LOL!
By the way, very expert meditators like Buddhist monks with tens of thousands of hours of formal sitting can maintain an absolute state of Samadhi outside of sitting if that's their desire. I'm pretty sure that's documented by EEG and fMRI findings carried out by neuroscientific research by the likes of Richie Davidson in his University of Wisconsin at Madison lab with Buddhist monks from Tibet.
The analogy he uses is that of a still lake vs. a turbulent one. That's not to say there's no brain activity but it's a distinct pattern and synchronized across widespread regions of the brain. It's that profound peace and serenity one experiences in deep meditation.
Practically speaking the end-result gives rise to a more pliable and flexible mind with clarity and creativity, love and compassion. Being able to relate to others on a more visceral level. A more discerning mind with a more intuitive grasp of right and wrong action and the free will to execute an appropriate course of action.
A very good YouTube video is "Becoming Conscious: The Science of Mindfulness".