The devil having dominion over water is some Mormon families tradition rather than actual doctrine or canon. The tradition could possibly stem from Joseph Smiths older brother Alvin drowning.
That the birthday party was on a Sunday does enlighten the situation for me. As stated already the degree to which families “Keep the Sabbath Holy” or adhere to any other teaching or doctrine varies, but many families would decline birthday party invitations on Sunday as a rule. If a childs birthday fell on Sunday, we would have a cake and small family dinner on Sunday but have the childs party on Saturday. Some families also would decline invitations to sleep overs on Saturday evening so all the family is home to wake up and get to worship services on time and together other families would even pick up children from weekend sports tournaments late on Saturday night, rather than let them be gone the entire weekend. Again, some families traditions and teachings are stricter interpretations of teachings and doctrine than others. I do not think that this family was offended so much as they just decided to decline without explaining. I did not always explain to casual friends why we might pass on a Sunday event. Rather than say, “Sorry we have church on Sunday.” Which can be taken as preachy or self righteous. I would just say, “We have prior commitments. Hope we can do something another time.”
That said, we regularly (monthly-ish) would invite another family over for dinner after church. These family dinners often felt similar to a party to me although they were family events rather than focus on one child. We would share a meal and play games. Watch the children perform their current piano or song practice and sit around and visit. So it isn’t that socializing on Sunday is forbidden, it is more that Sunday is considered the Lords day and family time.
Some families forbid even watching tv on Sunday others could watch tv if it was uplifting and some were watching their watches hoping to get home in time for kick off.
Some families would not let children play outside or ride bikes. We did allow those things in moderation and as long as it wasn’t when they were supposed to be in church or reaching out for Sunday calls to grandparents.
While we would not go boating on Sunday from home, if we were on vacation and saying on the lake, we would go boating after church then as long as we didn’t have to fill up the fuel on Sunday. When the fuel was out we were done until Monday morning. Not sure what the difference was to us, but that is how we did it. I suppose some families were stricter and others more lenient in their observance. My husband was raised in a different faith and did not feel that being out of doors or doing family things out of doors was breaking the Sabbath and felt strongly that it was possible to have reverent, spirit renewing and family strengthening experiences while outdoors.
Hiking, picnics, even a family ball game would all be appropriate after church activities in our family. As well as visiting the ill or elderly, calling or visiting extended family, reading, singing, playing music or family games. We tended to avoid household chores and yardwork on Sunday other than the normal cleaning up after yourself and meals sort of thing. If he had to work on Sunday, he did so without feeling bad about himself or his career choice because of it. (Airline industry.)
I was raised LDS (Mormon) from about age 8. And while my husband was not raised LDS we raised our children LDS. I am not currently an active participant and haven’t been for about four years, although I still attend to my grand childrens special days and events at church and when we are visiting them. We have not made a big deal about it. Just quietly stopped going unless someone special to us asks us to attend because it is a special event for them. If asked to identify religion as I was recently on an emergency room visit, I would still answer, LDS, even though I am not currently actively participating for myself.
I can see why you were concerned about offense, but I really doubt they were actually offended so much as just declined the party in favor of Sabbath observance traditions in their family. Then the usual busy life scheduling issues up to and after they moved away. I hope the girls get to get together and have a play date since they were so delighted to run into each other unexpectedly. Best of luck!