The fact that you are confused is not so strange in my opinion. Recently I spent some effort in this direction.
Definition of exogeneity/endogeneity in econometrics frequently is ambiguous. For this reason there is ambiguous treatment of the causality. Read here:
Regression and causality in econometrics
Note that endogenous/exogenous is a concept that should have only causal meaning. This point is matter of debate but my opinion is the previous. Read this related topic: Structural equation and causal model in economics
Other goal in econometrics is forecasting but in this setting the endogeneity problem do not play an important role. Read here:
Endogeneity in forecasting
Basically, the most important concept is that the exogeneity condition must related to structural error ($u$). Statistically speaking the most frequent definition is to mean conditional independence, like: $E[u|X]=0$ that is stronger than orthogonality $E[uX]=0$; note that $E[u]=0$ is valid by assumpion not by costruction. So the orthogonality and scorrelation between (structural) error and covariates/regressor are the same thing.
Note that in regression the orthogonality/scorrelation (now $u$ is regression error = residual), is valid by costruction not by assumption; in general $E[u|X]=0$ do not hold bu it is not very important.
Shortly, the interpretation of $u$ is crucial. Most confusion coming from this point.
Basically the so called "error term" tha you find in some econometrics presentation must be interpreted as true/structural error.
Others peculiarity about: orthogonality, correlation, conditional independence, full independence; can produce only confusion if the distinction between the two type of error above is not clear.