Contrary to what the current wiki page says, Simulacrum levels 3 and 4 are not just about ingroup signalling. See these posts and more, as well as Beaudrillard's original work if you're willing to read dense philosophy.

Here is an example where levels 3 and 4 don't relate to ingroups at all, which I think may be more illuminating than the classic "lion across the river" example:

Alice asks "Does this dress makes me look fat?" Bob says "No."

Depending on the simulacrum level of Bob's reply, he means:

1. "I believe that the dress does not make you look fat."
2. "I want you to believe that the dress does not make you look fat, probably because I want you to feel good about yourself."
3. "Niether you nor I are autistic truth-obsessed rationalists, and therefore I recognize that you did not ask me this question out of curiosity as to whether or not the dress makes you look fat. Instead, due to frequent use of simulacrum level 2 to respond to these sorts of queries in the past, a new social equilibrium has formed where this question and its answer are detached from object-level truth, instead serving as a signal that I care about your feelings. I do care about your feelings, so I play my part in the signalling ritual and answer 'No.'"
4. "Similar to 3, except I'm a sociopath and don't necessarily actually care about your feelings. Instead, I answer 'No' because I want you to believe that I care about your feelings."

Here are some potentially better definitions, of which the group association definitions are a clear special case:

1. Communication of object-level truth.

2. Optimization over the listener's belief that the speaker is communicating on simulacrum level 1, i.e. desire to make the listener believe what the listener says.

These are the standard old definitions. The transition from 1 to 2 is pretty straightforward. When I use 2, I want you to believe I'm using 1. This is not necessarily lying. It is more like Frankfurt's bullshit. I care about the effects of this belief on the listener, regardless of its underlying truth value. This is often (naively considered) prosocial, see this post for some examples.

Now, the transition from 2 to 3 is a bit tricky. Level 3 is a result of a social equilibrium that emerges after communication in that domain gets flooded by prosocial level 2. Eventually, everyone learns that these statements are not about object-level reality, so communication on levels 1 and 2 become futile. Instead, we have:

3. Signalling of some trait or bid associated with historical use of simulacrum level 2.

E.g. that Alice cares about Bob's feelings, in the case of the dress, or that I'm with the cool kids that don't cross the river, in the case of the lion. Another example: bids to hunt stag.

3 to 4 is analogous to 1 to 2.

4. Optimization over the listener's belief that the speaker is comminicating on simulacrum level 3, i.e. desire to make the listener believe that the speaker has the trait signalled by simulacrum level 3 communication (i.e. the trait that was historically associated with prosocial level 2 communication).

Like with the jump from 1 to 2, the jump from 3 to 4 has the quality of bullshit, not necessarily lies. Speaker intent matters here.