= Solution
Use the standard theorem on <reduction of torsion points on an elliptic curve>: at a <prime> of <good reduction>, reduction is injective on torsion whose order is <prime> to that <prime>. Reduction at $2$ implies that every odd-primary part of the rational <torsion subgroup> has order dividing $3$. Thus its only possible <odd prime> is $3$, and its $3$-primary part has order at most $3$. Reduction at $5$ injects the $2$-primary part into a group of order $9$, so that part is trivial. No other primary part can occur.
Consequently the <torsion subgroup> has order at most three. Since $P_1$ has exact order three,
$$
\boxed{E(\mathbb Q)_{\mathrm{tors}}=\{O,(0,0),(0,-1)\}\cong\mathbb Z/3\mathbb Z.}
$$
This argument uses only prime-to-residue-characteristic injectivity, and therefore avoids an unjustified claim of full torsion injectivity at the <prime> $2$.
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