Where did the spider come from?

I was intrigued to read Bob Woodward's book 'Rage' last week.  At the beginning he describes some background details in the lives of political characters caught up in the Trump story. He writes about  keen Christian friends, Mike Pence and Dan Coates together with their wives who met for a lengthy prayer session in their concern to follow God's guiding about their political futures. 

On this occasion Pence recounted the Old Testament story of David, who was hiding from King Saul in a cave when God sent a spider to weave a web across the cave opening. On seeing the web, Saul did not enter the cave. The spider had concealed David's presence and saved his life.  The story showed that even a spider might be an instrument of great salvation in the hand of God. 

Marsha Coates, whose grandparents were ministers, had never heard a sermon as serious and deep. The story raised obvious questions. Could a spider normally a cause for fear, bring salvation? (Page 25). 

You can guess why I was intrigued?  Two reasons (at least).  First, it's not in the Old Testament!  1 Sam. 24 certainly tells of David in a cave, but while he is hiding, the pursuing Saul comes into the same cave to relieve himself!  Oh, no spider?  David's men want him to kill Saul, but instead David cuts off a corner of Saul's robe, only to be conscience stricken and to confess to Saul leading to an extraordinary reconciliation.  

Second, where did the spider come from?  I immediately thought of the story I learned at school about Robert Bruce in the cave witnessing a spider at work and marvelling at its patience!  Surely not that! So, where did the spider come from?  Was it some misheard sermon or illustration that became confused with the bible story.  Did Bob Woodward misreport it?   Somewhere along the line biblical illiteracy is at work - all the more seriously if this is the most serious and deep sermon ever heard.