Yes, you’d need to use a trigger. (Just checking: is your posted trigger working correctly? At first glance, it looks fine to me.)
MySQL’s ON UPDATE CURRENT_TIMESTAMP is a pretty unique, single-purpose shortcut. It is what it is; this construct cannot be used similarly for any other values or for any column types other than TIMESTAMP. (Note how this functionality is defined on the TIMESTAMP type page instead of the CREATE TABLE page, as this functionality is specific to TIMESTAMP columns and not CREATE TABLE statements in general.) It’s also worth mentioning that while it’s specific to a TIMESTAMP type, SQLite doesn’t even have distinct date/time types.
As far as I know, no other RDBMS offers this shortcut in lieu of using an actual trigger. From what I’ve read, triggers must be used to accomplish this on MS SQL, SQLite, PostgreSQL, and Oracle.
One last note for passersby:
This is not to be confused with ON UPDATE clauses in relation to foreign key constraints. That’s something entirely different, which likely all RDBMSs that support foreign key constraints have (including both MySQL and SQLite).