Well spotted. This is a problem. After quite a bit of fiddling I found that the error is in Denis Simon's script used by Sage.
In fact, when executed with higher values of the parameters so that the script finds the rational points, it also prints the correct information.
The command
K.<t> = QuadraticField(-37)
E = EllipticCurve([K(17),K(0)])
E.simon_two_descent(lim1=10,lim3=50,limtriv=10, verbose=3)
prints correctly
[E(K):phi'(E'(K))] >= 2
#S^(phi')(E'/K) = 8
#III(E'/K)[phi'] <= 4
[E'(K):phi(E(K))] = 8
#S^(phi)(E/K) = 8
#III(E/K)[phi] = 1
#III(E/K)[2] <= 4
#E(K)[2] = 2
#E(K)/2E(K) >= 8
2 <= rank <= 4
When executed with lower parameter it produces the incorrect information
2 <= #III(E/K)[2] <= 8
I believe the script actually performs the correct calculations, but the part that draws these conclusions about the Tate-Shafarevich group has a bug in it. With the (correct) information that has been calculated before that line, the correct conclusion would be that $Ш(E/K)[2]$ has between $1$ and $2^3$ elements.
Magma is consistent. The following
K<b> := QuadraticField(-37);
A:=EllipticCurve([K!0,0,0,17,0]);
"Torsion::", TorsionSubgroup(A);
"Selmer Group::", TwoSelmerGroup(A);
"Rank::", Rank(A);
"Generators::", Generators(A);
T := A![0,0];
phi := TwoIsogeny(T);
"Phi-Selmer group ::", SelmerGroup(phi);
phihat := DualIsogeny(phi);
"Phi hat =Selmer group ::", SelmerGroup(phihat);
returns the information about the descent by isogenies and for the full $2$-Selmer group, rather than only the ones by isogenies.
Let $\varphi \colon E\to E'$ is the isogeny with kernel $\{O,(0,0)\}$. The $\phi$-Selmer group $\operatorname{Sel}^{\phi}(E/K)$ has dimension $3$ and so does the $\hat\phi$-Selmer group $\operatorname{Sel}^{\hat{\phi}}(E'/K)$. The $2$-Selmer group $\operatorname{Sel}^{2}(E/K)$ is also of dimension $3$, which means that the image of $\operatorname{Sel}^{\hat{\phi}}(E'/K) \to Ш(E'/K)/\phi(Ш(E/K))$ has dimension $2$. One finds that $Ш(E'/K)[\hat\varphi]$ is of order $2$, but the Tate-Shafarevich group of $E/K$ has a trivial $2$-part.
This is constistant with the Birch and Swinnerton-Dyer conjecture, I believe.
Best Answer
$\DeclareMathOperator{\sha}{Ш}$ I am not sure that the proof that Sha has order 9 is anywhere spelled out in full. Here the ideas how to do it.
First, that the order of $C$ is three in the $\sha$ is just saying that it has a point over a field of degree 3 (index=period), which is obvious, and none of degree 1, which was first proven by Selmer. See Cassel's lectures, chapter 18.
A $2$-descent, proves that $E(\mathbb{Q})$ has rank $0$ and that $\sha[2]$ is trivial. Calculating the torsion subgroup of $E$ (trivial), the Tamagawa numbers (all trivial and the value of $L(E,1)$, either using cm-theory or modular symbols, reveals that the Birch and Swinnerton-Dyer conjecture is equivalent to $\sha$ having $9$ elements.
Rubin's work on the main conjecture of elliptic curves with CM, as in "Tate-Shafarevich groups and $L$-functions of elliptic curves with complex multiplication" for instance, shows that the order of the $p$-primary part of $\sha$ is correctly predicted by BSD for all primes not dividing the order of the units in $\mathbb{Q}(\sqrt{-3})$. For our curve this means $\sha[p]=0$ for all primes $p>3$. But this does not cover the very bad prime $3$; though maybe has done this since, I don't know.
Selmer's original work proved that $\sha[3]$ contains $9$ elements, by doing a second descent $E\to E'$ via an isogeny of degree $3$. This exact example appears in magma's documentation of its ThreeDescent function: https://magma.maths.usyd.edu.au/magma/handbook/text/1515#17622 . More interestingly this calculation also shows that the $3$-Selmer group of the curves $E'$ at the other end of the $3$-isogeny is trivial. Hence the $3$-part of BSD is true for that curve and since this is invariant under isogeny, it is also true for ours. Therefore $\sha$ has order $9$ with the structure being $\mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}$.
This last step should also be possible using Heegner points with $D=-17$. Sage uses this in "prove_BSD" for the curve $E'$.