I am trying to understand the currency basis calculation and whether there is a difference in currency basis when quoted vs. OIS and -IBOR rates.
You can synthesise this with the single currency IBOR-OIS basis swap (SBS) in each currency.
For example paying the EUR/USD 10Y XCS @ -40bps, represents paying 3M Euribor -40 versus receiving 3M USD Libor flat. If you then buy a EUR 10Y OIS/IBOR SBS @ 8bps, this represents receiving 3M Euribor -8bps and paying EONIA flat. If you then sell a USD 10Y OIS/IBOR SBS @ 20bps, this represents paying 3M USD Libor - 20bps and receiving FFOIS flat.
Net you have synthesised paying a EUR/USD EONIA/FFOIS XCS @ -52bps (-40 +8 -20).
Note: due to the credit support annex (CSA) on the EUR SBS being EUR cash collateral through any clearing house this is different to that on the specific XCS instrument which (if it traded) would be USD collateralised on all aspects of the trade, so this will have a minor (probably negligible) impact.
Note: you can also use the same technique for other IBOR tenors such as 6M, where the standard XCS is 3M IBOR and there is a 3M/6M SBS market in each currency.