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I think what you wrote is correct and what the article says is correct. What you wrote is better and clearer than the article, which is unclear and garbled. Someone should fix this. The article is also garbled about "inextensible"; that needs fixing too. So it goes.
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433:{\displaystyle D^{+}({\mathcal {S}}):=\{p\in {\mathcal {M}}{\text{ such that }}\forall {\text{ past directed, non-spacelike curve from }}p{\text{ intersects }}{\mathcal {S}}\}}
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Maybe start with uninteresting(?) example: first 'everywhere at 1PM', next 'everywhere at local noon', then get more technical. (or, 1PM -: -->
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