I would write the equation of exchange as MV = Py.
The second term, Py is equal to nominal GDP. Assuming y is measured by real GDP, and that is calculated as Y/P, where Y is nominal GDP and P as measured by a price index, then y = Y/P implies that Y = Py.
So, the equation of exchange implies that Y = MV.
If M is the quantity of money and V is the number of times money is spent on final goods and services per period, then MV is spending on output per period. More money, or a change in how fast (veloclity?) it is spent on current output, might cause changes in nominal GDP.
On the other hand, if V is defined so that V = Py/M, then V is whatever it takes to make MV = Py hold. If M rises, then V falls. If P rises, V rises, and so on.
I think that one way to understand what those economists who use the equation of exchange as a theory have in mind simply requires that M be given two different subscripts. Ms for the quantity of money supplied and Md for quantity of money demanded.
The equation of exchange is Ms V = Py.
And V is defined such that V = Py/Md
By substitution, Ms *Py/Md = Py.
This implies that Ms/Md = 1
Or Ms = Md.
The equation of exchange is an economy theory based upon the quantity of money supplied being equal to the quantity of money demanded.
Now, if you define the quantity of money demanded as the amount of money people are actually holding and the quantity of money supplied as the amount of money that actually exists, and then add that someone is always holding whatever money that exists, then Md = Ms always.
I think that is what it means to treat velocity as being defined as V = Py/M.
Of course, what is usually done instead is to understand the demand for money to be the amount of money that people would like to hold. And further, that it is possible for people to actually be holding more or less money than they would like.
As an analogy, suppose quantity demanded is identified as the amount bought and quantity supplied as the amount sold. Since the amount sold is always equal to the amount bought by definition, quantity supplied and demanded are always equal. It is nonsensical, then, to claim that price depends on supply and demand, and in particular, the price adjusts so that quantity supplied and quantity demanded are equal. Since the amount bought and sold are always equal, it is not intelligible.
But, of course, quantity supplied is how much sellers would like to sell and quantity demanded is how much buyers would like to buy. Quantity supplied and demanded are not always equal. It is possible that the amount sold is less than the amount sellers would like to sell. Or that the amount bought is more than the amount buyers would like to buy.
Similarly, the amount of money people would like to hold may be more or less than the amount that they actually hold.
And that implies that velocity can differ from what people would like it to be. Does that sound odd and awkward? It does to me. And that is why I usually prefer to think about monetary economics in terms of the quantity of money and the demand to hold it, rather in terms of the equation of exchange. But still, I balk at claims that the equation of exchange is nothing but an identity. Because if it is, so is the notion that prices and real income adjust to bring the quantity of money demanded into equilibrium with the quantity of money supplied. And so is the notion that relative price adjusts to bring quantity supplied into equilibrium with quantity demanded.