In terms of writing guidelines I would write:
Statement A is true \citep{jon90}.
is printed as
"Statement A is true (Jones et al., 1990)" or
Statement A is true as shown by \citet{jon90}.
is printed as
"Statement A is true as shown by Jones et al. (1990)" or
Statement A is true according to \citet{jon90}.
is printed as
"Statement A is true according to Jones et al. (1990)" or
\citet{jon90} showed that statement A is true.
is printed as
"Jones et al. (1990) showed that statement A is true.".
In these 4 examples it would be grammatically wrong to replace \citet
by \citep
or vice versa. Depending on the natbib style the citations are printed slightly differently, but as far as I know they are always printed such that \citet
and \citep
stay grammatically correct independent of the natbib style.
\cite
however would be either grammatically wrong for the first of the 4 examples or for the last 3 of the 4 examples depending on the natbib style. (For some natbib style \cite
is equivalent to \citet
and for some \cite
is equivalent to \citep
.)
For more examples see https://tug.org/pracjourn/2007-2/garcia/garcia.pdf
For further variations and optional arguments see
https://www.overleaf.com/learn/latex/Natbib_citation_styles or
https://www.reddit.com/r/LaTeX/comments/5g9kn1/whats_the_difference_between_cite_citep_and_citep/, e.g.,
Statement A is true as shown in \citet[Theorem~1]{jon90}.
is printed as
"Statement A is true as shown in Jones et al. (1990, Theorem 1)" or