$B_n$ and $C_n$ define the same Coxeter diagram and the same Coxeter group

(The diagram and the leading minors.) The two names denote the same labelled path, hence the same Coxeter matrix, presented group and form [F1]. Let $C_k$ be its leading $k\times k$ submatrix and $d_k:=\det C_k$, with $d_0:=1$ and $d_1=1$. For $k\ge 2$ put $a:=\cos (\pi \mathrm{/}{m}_{k-1})$. The last row has only $-a$ and $1$ as potentially nonzero entries [F2]. Its diagonal cofactor is $d_{k-1}$; deleting row $k$ and column $k-1$ leaves a matrix whose last column has only the bottom entry $-a$, so a second Laplace expansion gives deleted-matrix determinant $-a{d}_{k-2}$. The off-diagonal cofactor has sign $-1$, and therefore $d_k=d_{k-1}-a^2{d}_{k-2}$ [F4], without assuming positivity. To compute the all-$3$ prefix, put $c:=\cos (\pi \mathrm{/}3)$: [F3] gives $2c^2-1=\cos (2\pi \mathrm{/}3)=-c$ and $c>-1$, hence $(2c-1)(c+1)=0$ and $c=1\mathrm{/}2$. Thus for $2\le k\le n-1$ the recurrence is $d_k=d_{k-1}-\frac{1}{4}{d}_{k-2}$. Induction, starting with $d_0=d_1=1$, gives $d_k=(k+1)\mathrm{/}{2}^k$ for $0\le k\le n-1$, since $\frac{k}{2^{k-1}}-\frac{k-1}{2^k}=\frac{k+1}{2^k}$ [F4]. At the final label-$4$ edge, [F3] gives $d_n=\frac{n}{2^{n-1}}-\frac{1}{2}\frac{n-1}{2^{n-2}}=2^{1-n}$. Consequently the leading minors of $2C$ are $2^k{d}_k=k+1$ for $1\le k<n$, and $\det (2C)=2^n{d}_n=2$ [F4]. This also covers $n=2$, using $d_0=1$.

(Small ranks and the parabolic $A_{n-1}$.) For $n=2$ the path has the single edge labelled $4$, which is the diagram $I_2(4)$, and this is the same labelled graph as $B_2=C_2=I_2(4)$ [F1, F6]. The subdiagram on $\{s_1,\dots ,s_{n-1}\}$ is the all-$3$ path $A_{n-1}$; hence $W_{A_{n-1}}=⟨s_1,\dots ,s_{n-1}⟩$ is a standard parabolic subgroup of $W$, isomorphic to the Coxeter group of type $A_{n-1}$, which is the symmetric group $S_n$ [F6]. Since the two names share the same labelled graph, type $C_n$ has the same Coxeter system and the same standard parabolic $A_{n-1}$, which is (iii).

(Positive definiteness, finiteness, and the same order.) By 1.1 [step 1.1] the leading principal minors of $2C$ are $2,3,\dots ,n$ and the full determinant is $2$, all positive; scaling by the positive factor $2$ does not change definiteness, so $C$ has all leading principal minors positive and is positive definite by Sylvester's criterion [F5]. By the finiteness criterion [F5] the group of $B_n$ is finite, and since $C_n$ has the same Coxeter matrix it has the same Coxeter system, the same form and the same finite group, in particular the same order [F1]. This is (i) and (ii).

(Conclusion.) The Coxeter matrices of $B_n$ and $C_n$ are equal, so the two names denote the same Coxeter diagram, the same Coxeter system and the same group, and the distinction between the $B$ and $C$ families lies in root-system data rather than in the presentation; the doubled cosine determinant is $\det (2C)(B_n)=2$ with leading minors $2,3,\dots ,n$, so $B_n$ and $C_n$ are positive definite and of finite type; and $B_2=C_2=I_2(4)$ while $B_n$ has $A_{n-1}=S_n$ as a standard parabolic, with $C_n$ identical.