I can answer for Welcome Home Wednesday your children are welcome to join you. It's a very laid back atmosphere, lots of families with kids. I haven't been to the AKL get together but would imagine kids are welcome there also.
There is an event on Wednesday's at Jambo house with a DVC guide. They call it the Sundown Mingle. They serve a few appetizers and an adult fruit punch. They hold the event on the viewing balconies right off the lobby. When we went in December, it was very low key and I don't even think there was any mention of sales or current offers. It was nice that they offered free childcare, which was done right in the lobby right next to the DVC kiosk.
As others have said, the Welcome Home Wednesdays is a family affair.
The Sundown Mingle is an adults event. But.... I have 3 kids, and they would only take the ones 3 and older for the childcare, so we had our 2 year old in the stroller the whole time. She did fine for the whole event.
The Sundown Mingle is fantastic. I think all DVC resorts would benefit from this. It made you feel connected and appreciated as a member, not like the Welcome Home Wednesday affair at the BW which I think is a disaster. I mean a snack ticket for a banana? Really?
Sorry, after the Sundown Mingle it really rubbed us the wrong way.