It is not true.
They don't chose rates, the rates are dependent on the contracts between the Visa/Mastercard and the members (banks issuing and clearing the cards and transactions).
There may in fact be two conversions: one is from the original currency to USD/EUR, and the other from the USD/EUR to the target (your home) currency. If your home currency is EUR, and the original currency is EUR, there shouldn't be a conversion, but it may so be that there is. It happens if the clearing house or the issuer operate with Visa/MC in USD (or similarly, in EUR, and the original and target currency is USD). Then transactions going through the network will be converted back and forth. It is rare, but I've seen it happening.
The volume of operations is such that neither Visa nor Mastercard will "wait" for a favorable exchange rate - they set the exchange rate. The volume of transactions allows them to offset currencies so that in many cases there's no real currency exchange necessary at all. For example - $1M of purchases by Europeans in the US offset most of the EUR1M purchases of the Americans in Europe. Yet, they will all be charged currency exchange fees, even though most of the currency hasn't been exchanged at all. So there's no need to pick and chose exchange rate, there's plenty of profit for Visa/MC there as it is.