On the meaningful level - it seems like "the law of the excluded middle."

Tertium non datur ("a third is not given")

Let's say that a correlates with c.
b also correlates with c, but to an even greater extent.
when b is controlled for, a no longer correlates with c.
when a is controlled for, b still correlates with c.

The error: someone says "a causes c". — Hallucinogen

For example: "a man's looks (b) are a strong factor in attracting a woman (c)".
"personality (a) also attracts women".
>> turns out what people label as "personality" correlates strongly with a man's looks.
The error: "personality attracts women". — Hallucinogen

A causes B IFF
1. There's a correlation between A and B (check)
2. B doesn't cause B [to rule out reverse causation]
3. There's no C that causes both A and B [to rule out third party causation]
4. The correlation between A and B isn't a coincidence

In the example you gave of what women are attracted to, a man's looks causes both a "good" personality and women to be attracted i.e. this is a case of third party causation and is a well-recognized pitfall in causal reasoning.

One often-cited example is the apocryphal study that "proved" ice cream sales "caused" shark attacks. The ice cream sales correlated well with shark attacks of course. However, the real cause was the summer heat which made people buy more ice cream and also increased both the volume and frequency of swimming (in the oceans and seas).

If you control for the high temperatures of summer (control for a man's looks) the correlation between ice cream sales (personality) and shark attacks (attractive to women) disappear. The summer thermometer readings (a man's looks) correlate with both ice cream sales (personality) and shark attacks (attractive to women). Simply put, to infer that ice cream sales (personality) causes shark attacks (attractive to women) is to fail to identify the real cause viz. the summer heat (a man's looks).

This fallacy goes by the name the third cause fallacy aka ignoring a common cause fallacy.

https://en.wikipedia.org/wiki/Correlation_does_not_imply_causation

Let's say that a correlates with c.
b also correlates with c, but to an even greater extent.
when b is controlled for, a no longer correlates with c.
when a is controlled for, b still correlates with c. — Hallucinogen

In addition to correlation does not imply causation. And assuming you're reading that correlation causally by specifying a causal model:

a correlates with c; could go a->c or c->a
b correlates with c; could go b->c or c-> a
a is conditionally independent of c given b; that says something like a->b->c or a->b<-c or a<-b->c, all undirected paths from a to c go through b.
b is conditionally dependent upon c given a; that says there is at least one path in the causal diagram from b to c or from c to b that does not go through a.

It looks like you're referring to the condition of confusing a collider (a->b<-c has b as a collider of a and c) for a fork (a<-b->c has b as a fork of a and c) or a chain (a->b->c) maybe?

Looks->attractiveness
personality->attractiveness
looks->personality or personality->looks

if looks-> personality then attractiveness<-looks->personality->attractiveness
if personality->looks then attractiveness<-personality->looks->attractiveness

The first has looks as forking personality and attractiveness as well as a chain to attractiveness through personality, the second has personality forking looks and attractiveness with a chain from personality to attractiveness through looks. So you might also be referring to the idea that a system can have more than one pathway from a variable to another through other variables, so blocking off one path through controlling/conditioning doesn't block off all of 'em just in case there's more than one.

Whether someone's committing an error of causal reasoning depends on the adequacy of the causal model, though. You can't tell colliders from forks or chains through reason alone, or even correlation/data alone!

As is often the case a correlates with b but there is a hidden variable x that is unknown, such that
x -> a -> b
or
x -> b
or
x -> a and x -> b

These hidden variables or factors can be the cause of much confusion.