Are laws governing transactions of individuals and companies that cross international borders private international law?
The actual laws that govern such transactions are not private international law. However, private international law does have an impact on such transactions.
Private international law refers to situations in which there is a conflict between the laws of different countries. When there is such a conflict, it can become necessary to pick which country's law will be used. One example of a situation in which this becomes necessary is when there is a contract between parties that are in different countries. Of course, this is the sort of situation that can arise quite often with the sorts of transactions that you are asking about. Thus, private international law does have an impact on international transactions.
However, the actual laws that are passed are not private international law. Instead, private international law comes into play when companies or individuals in different countries have disputes with one another and settle the dispute with a private international agreement.