You typically don’t get “ownership rights” when you purchase a game on Steam. You’ll typically be purchasing a licence to play the game, which could be taken away at any point.
That is certainly what Valve thinks and writes in their TOS but if their store has a big button that says “BUY HALO” then courts may very well decide that you actually bought Halo.
And many countries have a strict legal definition of what buying means that cannot be overruled by some company’s TOS.
First off, they’re synonyms
https://www.merriam-webster.com/dictionary/purchase#synonyms
Now, I’m certainly no expert on the US legal system. It certainly seems silly if you could circumvent entire laws just by using synonyms but what do I know.
However I have been talking about other countries where that is not the case and where the language is not English.
So It really doesn’t matter whether it say “buy” or “purchase” in English when it’s “kaufen” in German or “acheter” in French.