Prefixes of Stanley's Catalan paths with odd returns to the $x$-axis -- standard version and skew Catalan-Stanley paths

Stanley considered Dyck paths where each maximal run of down-steps to the $x$-axis has odd length; they are also enumerated by (shifted) Catalan numbers. Prefixes of these combinatorial objects are enumerated using the kernel method. A more challenging version of skew Dyck paths combined with Stanley’s restriction is also considered.