[David Cooper]

But the inertial RESISTANCE not “compensated” that way is still present, and, similarly to what in the hammer throwing case (A), it causes an outward pull on adjacent inner half moon. That is a real CENTRIFUGAL FORCE, quite similar to the one exerted by the hammer on the wire´s end. It “forces” closer half of the moon to keep the common orbit, instead of a smaller one that would match with the stronger earth´s pull on closer “hemimoon”.

If we think about an object sitting near a black hole, the gravity pulls on the object, and it does so more strongly on the near side of that object than the far side, so a tension force appears within it (which might pull it apart in the process known as spaghettification). That tension force is not any kind of centrifugal force. That force will be opposed by an opposite force which is likewise not any kind of centrifugal force.

If the planet is going round the black hole rather than falling towards it, these opposed tension forces are still acting within it, but their cause is exactly the same as in the non-rotating case. If you want to call the gravitational force centripetal force, what are you going to call the tension force that acts in the same direction? Is it centripetal force too? Maybe it is - it's just the force being converted to a different from and transferred on, so yes, and that logically requires you to call the opposing tension force reactive centrifugal force.

However, all these forces are driven by direct gravitational pull and the rotation aspect is a complete irrelevance to them, so if you want to provide people with a real understanding of what's going on, you have a duty to avoid using words like centripetal and centrifugal in the explanation.

Furthermore, if you still want to assert that any tidal bulge is caused in some way by "centrifugal force" based on misnaming a force as reactive centrifugal force (when it's really just reacting to direct gravitational pull), you still aren't managing to pin that label to anything that builds a bulge, not least because it would be attempting to prevent a bulge appearing rather than doing anything to help form one.

[rmolnav]

I didn´t replied last post because the discussion continued on the thread "What is centrifugal force?", and most of what posted was about centrifugal force in general, not to its relation with tides. If interested, you can follow it there.
But on a recent post D.C. referred to tides too, and I replied to that respect:
"Many, you included, say only differential gravity can cause tidal bulges …
Let us imagine moon´s gravity were constant across the earth, maintaining moon´s total pull, and therefore actual distances and moon-earth “dancing” …
The “tendency” of earth revolving particles (both solid and water) not to change their velocity vectors (INERTIA) would cause two “tidal” bulges, but BOTH in the sense opposite to the moon (opposite to the centripetal force, that is, always parallel to line earth C.M. - barycenter - moon C.M.).
That implies that the sublunar “bulge” would actually be the opposite: earth radius decreases at that hemisphere …
That would be similar to what happens if, with our hands, we make a cap of tea on a table follow uniformly a circular path …
As I´ve said on "Why do we have two high tides a day?" thread many times, those centrifugal inertial "effects", added to what caused directly by the varying gravitational moon´s pull (inversely proportional to the square of the distance), is what causes the real tidal bulges !!
Remember what Einstein thought:
"Einstein warmed to the idea that the gravitational field of the rest of the Universe might explain centrifugal and other inertial forces resulting from acceleration".
Do you know better than Einstein ?? Or do you think that gravitational pull, acting as centripetal force (by the way, your "grey" area ...) doesn´t cause centripetal acceleration, and subsequently neither centrifugal forces nor other inertial "effects" are present ??
If so, please kindly give all of us your "reasons", instead of just saying "your imagined centrifugal effects don't exist in the gravity case"

[David Cooper]

If you want to bring Einstein into it, you should use relativity. Relativity shows that straight-line differential gravity provides the same explanation regardless of the amount of movement perpendicular to the direction of the gravitational pull, and that leaves no possible role for centrifugal force.