OBR. Defensive coach goes to the mound and changes pitcher, with batter Smith coming to the plate. Coach leaves the mound and goes back to the dugout. While warming up the new pitcher, the catcher breaks a strap on his mask and needs a replacement, so he goes to the dugout. Now the coach wants to go back to the mound to visit with the new pitcher.
Is this a second trip to the same pitcher with the same batter at bat?

No

First trip with new pitcher. Don't know why he would do that since he just got through talking to him. If he did have something to say he could have the catcher convey the message.

Read 8.06, italic comments.

Ain't in italics on the website, but how 'bout this [from 8.06]:

"If the coach goes to the mound and removes a pitcher and then the manager goes to the mound to talk with the new pitcher, that will constitute one trip to that new pitcher that inning."

A manager or coach is considered to have concluded his visit to the mound when he leaves the 18 foot circle surrounding the pitcher's rubber. If the manager or coach goes to the catcher or infielder and that player then goes to the mound or the pitcher comes to him at his position before there is an intervening play (a pitch or other play) that will be the same as the manager or coach going to the mound. Any attempt to evade or circumvent this rule by the manager or coach going to the catcher or an infielder and then that player going to the mound to confer with the pitcher shall constitute a trip to the mound.