Does the New York lewdness law Penal Law § 245.00 require that a lewd act in public is intended to be seen?

The "intent to be seen" phrase applies to an act in a private place.

The plain reading of the statute shows that it is not the defendant's subjective intent that controls but rather whether it would be objectively reasonable for the actions to be visible by others.

The Court of Appeals has addressed your question. While not offering an explicit definition of "public place," section 245.00 read in its entirety does suggest that mens rea in the form of intent to be observed, reckless disregard of observation, or intent to arouse or gratify sexual desires by means of public observation is not required by section 245.00 (a). The crime is defined as a lewd exposure or act "(a) in a public place, or (b) in private premises under circumstances in which [the actor] may readily be observed from either a public place or from other private premises, and with intent that he be so observed." The placement of the phrase "with intent that he be so observed" after subdivision (b) rather than at the start of the section indicates that intent to be observed is required only in cases involving private premises, not public places. The structure of the section thus tends to contradict respondents' claim that lewd conduct is not performed in a "public place" unless the actor intends to be observed or has some other culpable mental state regarding observation by others.