Is it just because I'm an American reading English-languages sources, and so there's always this defensive framing? Or did the Soviet Union also envision the potential conflict in this way?
Pretty much all of the Soviet planning we have access to assumes that NATO attacks first, yes. One of the better-documented "command post" exercises, ЗАПАД-77 (West-77) opens this way, for example. Other windows into their strategic outlook at different points of the Cold War, such as the successive editions (1962, 1963, 1968) of Sokolovsky's highly influential Military Strategy, reinforce this view. Sokolovsky centers much of his discussion on the assumption that NATO was ideologically unable not to be permanently hostile towards the Soviet Union; this was consistent with Marxist-Leninist orthodoxy and it seems to me that at least some in the planning establishment really did believe it. Statements we have from after 1991 by high-ranking members of the Soviet military reflect this; Sergei Akhromoyev's memoirs mention a feeling in the 1980s that the general balance in Europe was turning against the Soviets & that the US was pressuring them very aggressively both politically and economically, and Col.-Gen. Andrian Danilevich's candid interviews in the so-called Hines Report of 1993-95 indicate a strong consensus in the General Staff that while the Soviets had absolutely no intention of starting a war, they believed a war with NATO was quite possible and they were planning vigorously to not lose. Now, in their eyes the best way to not lose was to either preempt this attack (to include, in an extreme case, preemptive nuclear strikes on NATO nuclear delivery systems), or at least to absorb it and counterattack aggressively without delay. There's strong, strong influence from 1941 here: a couple of successive generations of Soviet officers felt that the absolute worst thing that could happen on their watch was a second surprise invasion of the USSR with the war then being fought on their soil. There's also doctrinal reasons why they believed, in the modern age, the offense was the more advantageous form of warfare, since it allowed one to more rapidly disaggregate the enemy's military system - his logistics, command nodes, and second-echelon forces, without which he simply could not fight effectively.