For the sake of clarification, your question is which operation removes page splits. Are you referring to page splits as the fragmentation (logically or physically ordered pages/extents?) that gets removed in your answer? If so, both rebuild and reorganize remove fragmentation. If you are referring to compaction (removal of page splits?), as referenced in your resource, both rebuild and reorganize compact the index pages to the specified fill factor.
From your reference:
Rebuilding Indexes
Rebuilding an index drops and re-creates the index. This removes fragmentation, reclaims disk space by compacting the pages based on the specified or existing fill factor setting, and reorders the index rows in contiguous pages.
Reorganizing Indexes
Reorganizing an index uses minimal system resources. It defragments the leaf level of clustered and nonclustered indexes on tables and views by physically reordering the leaf-level pages to match the logical, left to right, order of the leaf nodes. Reorganizing also compacts the index pages. Compaction is based on the existing fill factor value.
I'm confused because running the script provided, the dbcc command shows the same number of pages after each operation (no page splits).
Would you mind clarifying how a page split (define?) is handled in each operation?