The goal of my answer is only to provide recent references.
I warmly recommend these two bits of T. Y Lam's book [2]:

- §I.8, for examples where transvections fail to generate $SL_n(R)$
- the second to last paragraph of §VIII.12 for other interesting examples of rings $R$ satisfying $SL_2(R) = E_2(R)$ or its negative.

And also B. Magurn's latest article on generalized Euclidean group rings [4].

**Update.**

Here are newer references focussing on the instances of $SL_2(R) \neq E_2(R)$ for $R$ a quadratic order in a totally imaginary quadratic field. The state of the arts is to be found in [3] and [6], while [5] gives a nice geometric insight on the set $SL_2(R)/E_2(R)$.

An older, but in my humble opinion, important paper is [1], where the structure of $SL_2(R)$ as an amalgamated product with factor $E_2(R)$ is described for $R$ the ring of integers of a totally imaginary quadratic field (with few exceptions), see Theorem 2.4.

[1] C. Frohman and B. Fine, "Some amalgam structure for Bianchi groups", 1988.

[2] T. Lam, "Serre's problem on projective modules", 2006.

[3] B. Nica, "The unreasonable slightness of $E_2(R)$ over imaginary quadratic rings", 2011.

[4] B. Magurn, "On a note from Oliver concerning generalized Euclidean group rings", 2014.

[5] K. Stange, "Visualizing the Arithmetic of Imaginary Quadratic Fields", 2017.

[6] A. Sheydvasser, "A Corrigendum to Unreasonable Slightness", 2017.