I remember reading that Lebanon had been experiencing a dual exchange rate phenomenon. In which the official one is much less than the market rate. This has happened in other countries like Argentina and Venezuela before too

If this were the case. The benefits of using a foreign card would depend on the exchange rate in which card payments are settled.

Are Lebanon's card transactions with foreign cards settled at the official or at the market exchange rate? Is the difference between both rates still significant?

Yes, because they go over a common network, Whether they can be used in Lebanon is different from whether they issue cards to citizens in Lebanon