And there are about 100 more to consider as directed in the constitution.
So you do not agree that the constitution says that making treaties requires the concurrence of two thirds of the Senate?
Sure it does. It doesn't say that a treaty is the only form of international agreement which can be entered into. A POTUS can enter into any kind of international agreement besides a treaty if either a) a statute requires or explicitly permits him to do so, or 2) he is exercising his powers and duties as delegated to him by the Constitution, such as his power to represent the United States in foreign affairs, or his power as commander-in-chief of the armed forces.
This has been long settled, since all the way back at the beginning, and is not debated by...pretty much anyone. But hey, you're entitled to have your own issue with it now in 2017. You won't get anywhere with it, but argue away.
If you're wondering what the difference is between a treaty and an international agreement besides how they are adopted, the most important distinction is that a POTUS probably can't unilaterally pull out of a treaty like Trump just did with the Paris accord. Had it been a treaty, he likely would have needed Congressional consent to pull out. Had Obama been able to ratify this as a treaty - if he had the votes he needed - I'm sure he would have done so, because it would have made it very difficult for a successor in office to do away with it.