It does depend on definitions.I suppose that is true if one were to insist that original sin directly transformed mans nature to be a "sin nature". But the idea that man had a sin nature is easily found in scripture from the description of man at the time of Noah, the description of men in the behavior of God's own people, the description of men by Paul in Romans 1, and so on. This seemed to occur very early, be universal and be in some way remedied by the Holy Spirit. Other than that I guess that is why you have all the different theologies by truly serious men. So man got it somewhere, it is universal, and seems to be the case with all men of all races. The thing we have in common is our head, Adam. And we know something very pivotal happened with him.
Let's clarify -
Scripture refers to our natural state as "flesh" and as opposed to "spirit".
If we used "flesh" and "spirit" then perhaps it would be easier.
By "sin nature" if you mean a moral nature that is inclined to the flesh then I agree. It just seems easier to call this a "human nature" as nature itself does not equate to sin but temptation (see James 1-3).
This is how Christ could be the Son of Man (or "Adam"), btw. He was tempted in all points as are we (he had this human nature....you'd say "sin nature") but without sin (although tempted He did not give in to that temptation).