No, neither definition is correct.
Internal fragmentation: space free on pages within the index. Shown by avg_page_space_used_in_percent in sys.dm_db_index_physical_stats.
External fragmentation: pages out of order within the index (logical order doesn't match physical). Shown by avg_fragmentation_in_percent in sys.dm_db_index_physical_stats.
Rather don't use the terms 'internal fragmentation' and 'external fragmentation' as they have no precise definition and everyone means something different.
Logical fragmentation - pages out of order
Low page density - space free on pages.
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