in favour of purple martin ID:
grayish throat (esp in photo "3 of 58", which is the only one not overexposed) and fine vertical streaking on breast should rule out tree swallow
purplish blue on top of head - juv. tree swallow has no blue/purple at all
rather stout bill - tree swallow's is more petite
nothing to judge relative size really, but this bird still looks too large and thick, also rather long-toed
pose - hanging out in top of tree with dead branches is exactly what purple martins do - I saw dozens of these in Louisiana last week looking and posing exactly like this
lakeshore location obviously does not rule out swallows but is perfect for martin