No, a "corrections trade instructor" would not be considered a "Public Safety Employee" for tax purposes.
From the IRS: For purposes of § 72(t)(10), the term “qualified
public safety employee” means an employee of a State or of a political
subdivision of a State (such as a county or city) whose principal duties include
services requiring specialized training in the area of police protection,
firefighting services, or emergency medical services for any area within the
jurisdiction of the State or the political subdivision of the State.
The best way to understand this is that "public safety employees" are those whose duties are "outward facing", i.e. directed towards the general public. While your duties certainly serve the "public interest", you don't have any direct contact with them in the process.