I work in a preschool and this is really common I think you sum it up, all the possibilities....being too engrossed in play and leaving it too late, being shy to ask to go, the unfamiliar toilet/seat/potty. It just sometimes takes them longer at nursery than at home. The nursery staff will have seen this before and certainly won't disbelieve you and how well he's doing at home. Just keep on as you are and he'll get there in time
DS is 2y5m and decided last Weds that he didn't want to wear nappies any more. It wasn't exactly great timing as DD was born 3 weeks ago but tbh he has been showing signs of being "ready" for a while so didn't want to force him to stay in nappies.
Anyway from day 1 he has "got it" at home and we've had one single tiny wee accident in 6 days. We've also been out to cafes, on buses etc without any accidents.
We don't do prompting, he just takes himself to the toilet when he needs to and let's us know when he is done or needs his bum wiped. Due to the weather he has mostly been naked or bare bummed at least so no issues with getting pants off etc.
Nursery though is the opposite! 3 wee accidents every day he has been there - and one poo accident too. I don't know if it's the feel of pants on him, too busy playing, shy to ask the staff or what. We sent his own loo seat from home in today to see if it would help but no difference. I know it's early days but just seems odd to have such a discrepancy and tbh it's a bit embarrassing - I'm sure the nursery must think we are joking about him having no accidents at home!