I am a bit confused by various references to G-d’s presence (Shechinah) resting at one place or another. Given that G-d is present everywhere, all the time and always nearby, I am not sure exactly what is meant by these references. I am having trouble formulating this question. I hope this is clear.

Hashem is everywhere all the time – however WE can't see that, Hashem appears to us in different times more than others and in different places more than others, we relate to that notion as Shechina, the presence of Hashem. Therefore it's not Hashem that is more in one place but we that can find more godliness in one place more than another This is a very simple way of explaining – there is much more to it as you get deeper and deeper into Kabbala.