Well, yes. But a line has to be drawn between a low-power agreement (like: I won't tax your wheat if you don't tax my steel) and a high-power agreement like the TPP. I would argue that that line should be drawn as conservatively as possible.
If such an agreement has sweeping powers, or the potential for such powers, it's a treaty.
It’s a treaty when Congress or the President want it to be. The Constitution is not specific.
You might be interested in a little history of treaties vs. agreements:
http://constitution.findlaw.com/article2/annotation12.html#f388