Why is this "the first elliptic curve in nature"?

I actually only wrote the part that says that this curve is a model for $X_1(11)$, not the first part, which I think was written by John Cremona.

It is standard to order elliptic curves by conductor (e.g. for statistics), and 11 is the smallest possible conductor. However, there are 3 curves with conductor 11, and no canonical way to order them as far as I know (though @François Brunault has an interesting point); for instance LMFDB labels do not order these 3 curves in the same way as Cremona labels.

This curve being the first one could maybe also be understood in terms of modular degree, although this is also ambiguous: if we order them by degree of parametrisation by $X_1(N)$, then this curve, being a model of $X_1(11)$, comes first, but if we order in terms of degree of parametrisation by $X_0(N)$, then 11.a2 comes first since it is a model for $X_0(11)$.

I can only echo Tim D's explanation: from Coates via Vlad to me. I did not know about it having minimal Faltings height.

The closest thing I found in Diophantus is problem IV(24) which is solving the system $$X_1+X_2=a,\quad X_1X_2=Y^3-Y.$$ Diophantus sets $X_1=x$ and eliminates $X_2$ obtaining $$x(a-x)=Y^3-Y.$$ This seems to be the first elliptic curve encountered in the book of Diophantus; before that he only considers rational curves and surfaces.

Diophantus choses $a=6$ and obtains a solution $x=26/27,\; Y=17/19$.

(This little research is based on a Russian translation of Diophantus with comprehensive comments by I. G. Bashmakova, published in Moscow in 1974.)