My card AID ends in 31010, yet my bank used to issue me with a Solo and was told at the time of conversion that it'd work exactly the same as it, yet I haven't purchased a ticket from a guard as I live in London which is a penalty fares zone, so can't confirm it it works offline or not.
Also, I thought part of the number on the card confirms if the Visa works as an ex-Solo or not, which may also stop offline transactions?
If it makes any difference, if I purchase from a London Overground ticket office (and the vast majority of retailers), the receipt will say "VISADEBIT" yet from LUL it's "VISA DELTA".
If it's the blue Natwest/RBS one then it is unlikely to work offline. Just because RBS/Natwest have changed their card systems from Solo/Maestro > Visa it is unlikely to change your type of card unless the bank considers you to be of less risk or your account type changes.
Since the removal of the distinction between Visa and Visa Electron (which had different AIDs - Visa Credit/Debit being 31010 and Electron being 32010 all of the 'rules' for allowing online and offline transactions are now stored on the chip and magstripe and terminal config. It's not just the AID alone that will cause the card to be declined when working offline, there are a load of other factors too...
It's a little harder to detect the differences between offline and full online cards just by looking - but in general cards which allow offline auth have more features, such as cheque guarantee (no longer used but it's a reasonable benchmark) or contactless.
The thing you mention with "VISADEBIT" is just purely cosmetic. A lot of retailers choose to use what is called the "Application Label" off the card to print on the receipt - that is, the cosmetic name that the issuer of the card has chosen to give the Visa Debit application on the card.
In Natwest's case, they forgot to press the space bar when naming it.
HSBC's label for Visa Debit is just 'Visa', whilst others choose to write it all in capitals or in a normal format.
Obviously in LUL's case they have chosen to use the names that are pre loaded into their system. "Delta" hasn't been around for ages (branding-wise) and it shows how old the systems may be...