Obviously, there is no official definition, but in general you’d say “I am hedged” if you have somehow reduced your variance without a significant decline in your alpha. Since in most spread trades you would not be able to clearly distinguish which one is the alpha leg, it’s hard to make that statement.
Let’s take your example. Take a stat arb trading SPY/QQQ spread vs a long/short PM that is buying Space Sprockets Builders International (SSBI) and selling some index to short. The latter knows everything there is to know about the company (or so he thinks) and simply want to reduce his outright market exposure. The former initiated the trade because the spread is dislocated and he is actually interested in exposure to the spread as an asset. If a stat arb PM (eg yours truly) is trading a bunch of event stocks and is shorting some spooz against them, he is hedging, not buying a spread.
PS. Feels like the “what’s in a name” type of discussion - I’d say we are both right