Quite right, Monte. A person doesn't have to just flip a coin or "do what feels right." The API energy conserving logo is the motorcycle equivalent of "leaves of three, let it be" for avoiding poison ivy.
For every current Triumph engine up through and including the Rocket III, the spec is any semi or fully synthetic 10W/40 or 15W/50 motorcycle* engine oil meeting API SH or higher
and JASO MA specs. (*The 'motorcycle' designation, as a practical matter, means only without those friction additives.) Any oil meeting those specs ought to be fine.
Even if it doesn't cost $60/gallon.