So far on this thread the facts stated all appear correct - the opinions and interpretations I'm less sure about.
If you are VFR in Class G and a controller (whether in uniform or not) asks you if you could please do x or y, you can say yes and do it or say you'd rather not, thank you. If you say you'll do it please keep your side of the bargain or tell the controller that you've changed your mind, and if you say no it's always nice to give a brief explanation why.
Be careful of making assumptions. If the controller asks you to turn it's not necessarily to identify you (although I admit there's not much other reason for the request). If you are not told that you are under a radar service ... then you're not getting any sort of radar service.
Speaking as a controller, taking big aeroplanes through Class G is an interesting task and involves careful interpretation of the rules if everyone is going to get the best deal that they can. No separation is required twixt VFR and IFR but the big aeroplane is often under a service where the controller has to make sure it stays away from other radar targets (VFR, IFR , talking to the controller or not talking, day or night ... you get the picture). But there's a balance to be found. If the controller tries to identify every aircraft that talks to him/her, the workload increases significantly for relatively little benefit.
FWIW using fly not above etc. if the aircraft is not identified sounds a bit iffy.