Beyond the rhetorical (orator), was he influential in deciding troop movement, placement, attack strategies?
Trotsky's chief value has more to do with is organizational and motivational skills versus his ability as a strategist. In fact his reforms to the Red Army were probably his greatest achievement in the Bolshevik cause.
After the October Revolution, Bolshevik power stemmed from the units of Red Guards spread throughout the country, mostly in cities. Unsurprisingly, these ad-hoc formations proved ill-disciplined and totally unsuited for warfare, especially when they came up against regular soldiers like Denikin's Volunteer Army or the Germans. In March 1918 Trotsky became War Commissar, and implemented sweeping reforms of the Red Army.
The most important one, probably, was that Trotsky recognized the Red Army would need Tsarist officers if it was going to be led efficiently. The Bolsheviks called for volunteers in March and then in the summer began drafting former Tsarist officers (sometimes at gunpoint). Nevertheless, these "military experts" professionalized the Red Army and made it into a force that could stand up to its enemies in a real battle.
Trotsky's other big accomplishment was boosting morale. Despite representations of the Russian Civil War as a popularity contest between Reds and Whites, most soldiers on either side were peasant conscripts who had little stake in the political battle. Trotsky did a very good job at keeping the Red Army together, through a combination of political education, morale-boosting and ruthless discipline. He spent most of the war traveling across Russia on his armored train, instilling political enthusiasm in Red armies. He appointed political commissars as watchdogs and educators, targeted young and enthusiastic communists as recruits, and used propaganda to tell his soldiers why they were fighting. If that didn't work, he also used brutal repression to prevent disobedience. Tactically-speaking, we have his famous "blocking detachments" detailed to shoot any soldiers who retreated from battle. He was also a firm proponent of terror as a way to entrench the Bolshevik regime.
Tl;dr - Trotsky's importance was on the bigger, strategic level as an organizer, propagandist, and disciplinarian. His reforms allowed the Red Army to become a cohesive force capable of taking on its opponents in battle. I hope that answers your question - you seem to be asking more if Trotsky was a valuable tactician, but to the best of my knowledge that was not his value. However, he allowed the right tacticians to rise to the fore.
Sources:
Mark Hagen - Soldiers in the Proletarian Dictatorship: The Red Army and the Soviet Socialist State, 1917-1930
Richard Pipes - Russia Under the Bolshevik Regime
Bruce Lincoln - Red Victory: A History of the Russian Civil War