When soldiers of country X are tasked
* with killing soldiers of country Y, there is one generally accepted word for it: War, between X and Y.
In contrast, when country X "only" supports forces engaged in a war with country Y, country X is not considered at war with Y.
Let's take some examples:
Russian and China support North Vietnam, fighting the USA. At war: North Vietnam vs USA + South Vietnam.
The USA + Pakistan support the Afghan rebellion against the USSR. At war: USSR vs Afghanistan rebels
South Korea, first phase.
Russia and China support North Korea vs South Korea and USA. At war: N Korea vs S Korea + USA.
South Korea, second phase.
China intervenes in North Korea vs South Korea and USA. The "war" state vs China is problematic and intentionally kept limited by the US, but Mac Arthur does not see it that way and loses his job for it.
Note that 1950s China is not nuclear which probably contributed to Mac Arthur's rather cavalier attitude.
At war: N Korea and arguably China vs. S Korea + USA.
Bottom line: there is a huge difference, historically, and that includes Cold War conflicts involving both Russia and the USA, between being an active belligerent and providing support. Both NATO and the Warsaw Pact took great care not to put their troops into direct conflict, even though both at times provided ample material support against the other.
Let's keep it that way.
* as opposed to cases where either there are accidental deaths or there is a localized, unplanned, incident leading to deaths.