There may be the odd reference to Arriva plc knocking about if they have not bothered to formally re-register the name at Companies House as plain Arriva, but the document you linked to and other parts of the website are very careful to just talk about Arriva. If you can't buy its shares on the stock market, it isn't a plc.
I don't think that's entirely correct. The difference between a Limited company, and a Public Limited Company is that a PLC has no way of controlling who owns its shares. A limited company can simply decline to sell shares to someone. Whilst PLCs are usually listed companies, they are not always, either through takeover or where PLC has been chosen as the appropriate legal status.
In the case of Arriva, it seems to me (and this is usual) that Arriva PLC continues to be a PLC, however its shares are all in the ownership of DB. Therefore its has been delisted from the stock exchange, and remains a PLC, even though you and I cannot buy shares due to them all being owned by DB.
There are very few companies who do not have to display their legal staus (LTD, PLC, LLP and so on) these are usually state owned businesses or those set up by specific legislation.