This is totally right, you absolutely want to sugar coat it. Don't say "I'm leaving because I hate bill". But for everyone else's benefit you can drop in that this new place you're working for gives you the flexible hours you require and is paying you more money.
The point was more, "if you're unhappy, find something new." You don't owe anyone anything, and if they can't figure out how to incentivize you into staying, you shouldn't. It sounds like old manager had a good grip on it and new manager does not, in this particular case.
Yup, important communication is often best when informal. For forms, official interviews and the like, carefully protect your goals. But be creative in considering how you can educate the other party.
The point was more, "if you're unhappy, find something new." You don't owe anyone anything, and if they can't figure out how to incentivize you into staying, you shouldn't. It sounds like old manager had a good grip on it and new manager does not, in this particular case.