Understanding the effects of government deficit on international trade
No, your logic doesn't make a ton of sense. That doesn't mean that the Trump tax cuts are a good thing.

Foreign debt isn't the same as a country removing money from the economy. The way the U.S primarily finances debt is through the sale of treasury bonds (T-bills). Essentially an agreement with a private individual (sometimes foreign) that over the course of X time period the debt will be paid back with interest. 

This is the same as making a loan with your local bank. They give you money, in this case $5 trillion and after a period of time you pay off the principal loan plus interest payments. The assumption that Trump (and all American presidents since Bill Clinton when we last had a current account surplus) is making is that in the period between making the loan and paying it off the economic stimulus from the tax cuts will outweigh the loan and its payment. The U.S gets extremely favorable terms on its loans with virtually 0 interest rates meaning that the 5 trillion stimulus in decreased taxes may very will increase tax revenue through increased economic activity by more than 5 trillion dollars which is the argument that Trump is making.

Whether or not this occurs depends on where we are on the Laffer Curve which essentially states that at a certain tax rate decreasing taxes will actually increase government revenue by stimulating businesses. Personally I, and many contemporary economists believe that we are not at that point on the Laffer curve and a tax decrease would in the medium run lower output, however clearly Trump disagrees and there is no true consensus among economists as to where we currently fall on the Laffer Curve.