Let's assume for now that the item is in a circular orbit (just to keep things simple)
The Gravitational Pull is counterbalanced by the Centripetal Force
This sounds to me like a case of the anti-centrifugal brigage "correcting" a statement by replacing the word "centrifugal" with the word "centripetal" and, in doing so, turning the statement into nonsense.
We can look at the orbit problem from two reference frames.
First let's look at things from a rotating reference frame that rotates with the satellite's orbit. Since this is a rotating reference frame we have a centrifugal force and we can make the statement,
- The Gravitational Pull is counterbalanced by the Centrifugal Force.
Now let's look at things from a non-rotating frame.
- To keep something moving in a circle, a centripetal force is needed. In the case of an orbit that, centripetal force is provided by gravity.
It's important to realise that these are just two ways of describing the same situation from different reference frames.
However this has little to do with "weightlessness". The key to understanding weightlessness is that we don't feel gravity or acceleration (and centrifugal force is just another way of looking at acceleration) directly. When we talk about feeling our weight what we are really feeling is not gravity but the contact forces, tensions and compressions on/in our body that are acting against gravity. When we talk about experiencing "G-forces" what we are really feeling is not the acceleration but the contact forces, tensions and compressions on/in our body that are causing the acceleration.
If those contact forces (and therefore the resulting tensions/compressions) are negligible* we don't feel gravity. This applies equally to being in orbit, the NASA Vomit Comet or the early stages of skydiving (before your body has accelerated enough for air resistance to be non-negligible).
Orbit is interesting mostly because it can be sustained for much longer periods of time then the other weightlessness scenarios.
*They aren't actually zero in a satellite because the gravitational field is not quite uniform across the satellite and because satellites in low orbit do experience some air resistance.