IMO, if you say you went to a shop 2 times, then that is it, you didn;t go before and you'll not go again.
If you went to the shop twice, you could always go again and you may have been before.
But that's just how I see it.
also - it's unlikely nowadays to say I went thrice - seems normal to say three, four, five times etc yet equally normal to say I went once, twice. No parity in the rule is there?
Chambers Dictionary defines "twice" as follows, "adverb 1 two times � Twice two is four. 2 on two occasions. 3 double in amount or quantity � twice as much."
I would say twice as it can mean on two occasions.