I think the "my kids come first" saying is the death knell in any second marriage. It sets up a "some are more equal than others" Orwellian mindset that just isn't healthy. No one should come first and as other posters have said, the relationship/love we have for our kids is innate, unconditional and different.
I have a problem with Anna Pasterkak and this article. Firstly (and I'm not judging as I don't know the situation fully) for these 3 daughters, their father was still married to their mum. Whilst that long term relationship may have needed some work, their dad chose to enter into an affair. It doesn't surprise me therefore that the daughters were hostile to the new wife and two of them refused to meet her. The couple also met, got engaged and married very quickly and there seemed little interim for the situation to sink in and settle. The way she describes him talking to his daughters is the way he most likely always talked to his daughters. Why should he change that? My DH calls me 'darling', he also calls his daughters by the same endearment. It's a term of affection. No problem with that. It seems to me that Ms Pasternak wants to be his one and only darling and the problem lies with her and her insecurity. Not him, nor his daughters.
As for writing a thank you note after visiting, no, not really. Wouldn't she rather feel that his kids visited their "second" home in a state of relaxed, "combined family" atmosphere rather than as an invited "guest"?
My situation is that I met my DH 3 yrs after his ex-wife's affair, which ended their marriage. We went out for 4 years before I moved in and were married a month later. His (now adult) kids felt it was ok for dad to have a GF but NOT to marry her and I've been treated as "the other woman" for such a long time now that I have completely detached from them at times. When they are Ok with me, I am very happy to see them, ask of them, speak to them. When I'm ignored, I treat them with detachment. This may seem cold but we're nearly 9 yrs on now and it is, for me, a survival mechanism. Given my time again, I'm not sure I'd have tried so desperately hard to welcome them and for them to like me. The more I tried, the more they didn't like me. The more their dad loved me, the more they resented me. But that's a whole other subject.
Daily Mail. I so dislike that "news paper".