doc: update spin lmp doc and example #10168

Job	Run time
Test Python (1, 3.9)	28m 14s
Test Python (1, 3.12)	39m 17s
Test Python (2, 3.9)	28m 3s
Test Python (2, 3.12)	35m 34s
Test Python (3, 3.9)	28m 30s
Test Python (3, 3.12)	39m 0s
Test Python (4, 3.9)	28m 12s
Test Python (4, 3.12)	36m 58s
Test Python (5, 3.9)	27m 47s
Test Python (5, 3.12)	37m 15s
Test Python (6, 3.9)	29m 6s
Test Python (6, 3.12)	37m 38s
Combine and update integration test durations (3.9)	11s
Combine and update integration test durations (3.12)	9s
Pass testing Python	6s
	6h 36m 0s

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