First, neither of these equations that you have discussed here is in the correct form for a cause-effect relation. In a cause-effect relationship, by definition, the cause precedes the effect. So causal relationships are given by equations like this one $$B(t)=\int_{-\infty}^{t} A(t_r) dt_r$$ or even $$ B(t)=A(t_r) $$ where $A$ is the cause and $B$ is the effect and $t > t_r$. These equations are not symmetric; the cause and the effect are unambiguously identified by the fact that the cause is earlier than the effect.

In electromagnetism, for example, Maxwell’s equations are in a non-causal form, while Jefimenko’s equations and the retarded potentials are in a causal form. Although commonly stated, it is incorrect to describe Maxwell’s as saying that a changing E field causes a curling B field etc. It is correct to describe Jefimenko’s equations as saying that charges and currents cause E and B fields.

When I searched on the Internet, I saw that it is usually interpreted in a cause and result way: energy and momentum cause a curvature in space-time.

It may be common, but it is an incorrect interpretation. Causes precede effects. In that equation the energy and momentum do not precede the curvature. So Einstein field equation (EFE) does not support that claim.

The ADM formalism is closer to a causal equation for GR than the EFE.

Nowhere I see that it is interpreted as an equivalence, so that it can also go in the other direction: a certain curvature in space-time will cause energy-momentum

This is, in fact, common in the literature. Not in the incorrect sense that you have stated here where curvature causes stress-energy, but in the correct sense that curvature implies stress-energy.

For one specific example, in the derivation of the Alcubierre spacetime, the metric was stated first and the implied stress-energy tensor was then derived from that. Many other spacetimes were derived using that metric-based approach, which is founded on the fact that the equivalence relationship in the EFE does go both ways.

First, neither of these equations that you have discussed here is in the correct form for a cause-effect relation. In a cause-effect relationship, by definition, the cause precedes the effect. So causal relationships are given by equations like this one $$B(t)=\int_{-\infty}^{t} A(t_r) dt_r$$ or even $$ B(t)=A(t_r) $$ where $A$ is the cause and $B$ is the effect and $t > t_r$. These equations are not symmetric; the cause and the effect are unambiguously identified by the fact that the cause is earlier than the effect.

In electromagnetism, for example, Maxwell’s equations are in a non-causal form, while Jefimenko’s equations and the retarded potentials are in a causal form. Although commonly stated, it is incorrect to describe Maxwell’s as saying that a changing E field causes a curling B field etc. It is correct to describe Jefimenko’s equations as saying that charges and currents cause E and B fields.

When I searched on the Internet, I saw that it is usually interpreted in a cause and result way: energy and momentum cause a curvature in space-time.

It may be common, but it is an incorrect interpretation. Causes precede effects. In that equation the energy and momentum do not precede the curvature. So Einstein field equation (EFE) does not support that claim.

The ADM formalism is closer to a causal equation for GR than the EFE.

Nowhere I see that it is interpreted as an equivalence, so that it can also go in the other direction: a certain curvature in space-time will cause energy-momentum

This is, in fact, common in the literature. Not in the incorrect sense that you have stated here where curvature causes stress-energy, but in the correct sense that curvature implies stress-energy.

For one specific example, in the derivation of the Alcubierre spacetime, the metric was stated first and the implied stress-energy tensor was then derived from that. Many other spacetimes were derived using that metric-based approach, which is founded on the fact that the equivalence relationship in the EFE does go both ways. So this equivalence is, in fact, both well known and commonly used.