One of the first things discussed in books on elliptic curves is the change of variables that "puts the curve" in Weierstrass form. To give a concrete example, the curve $6y^2=2x^3+3x^2+x$ (the "cannonball problem" discussed at the beginning of L. Washington's book) can be put in Weierstrass form $v^2=u^3-\frac{9}{4}u$ by the change of variable $(u,v)=(3x+\frac{3}{2},9y)$. In fact, the explicit formulas to define the group law is usually only given for the Weierstrass form, with the understanding that one can always reduce the more general form to it (assuming the characteristic of the field is not 2 or 3). My question is about the precise meaning of "puts the curve" above. The two curves are surely not "isomorphic", as defined for example in http://www.lmfdb.org/knowledge/show/ec.isomorphism
I understand that of course studying and deriving properties of the simpler curve $v^2=u^3-\frac{9}{4}u$ one can easily go back to the original curve $6y^2=2x^3+3x^2+x$ using the simple transformation, but it seems to me that a discussion about the relationship between the two curves would be helpful. But I did not find it in any of the standard references on elliptic curves I know. To put my question in simple minded terms, what happens to the elliptic curve when we make a change of variables? Because one thinks of two curves as being "the same" if they are isomorphic, but apparently the first thing one does when studying an elliptic curve is to change it to another one that is not "the same".