Not sure we can fault WTBTS for this teaching as there is Biblical precedence for it. I haven't done a deep study into this particular aspect of the covenants (their legalities), however, they are probably basing these "legal contracts" on the language of Hebrews 6:16,17:
For men swear by someone greater, and their oath is the end of every dispute, since it is a legal guarantee to them. 17 In this same way, when God decided to demonstrate more clearly to the heirs of the promise the unchangeableness of his purpose, he guaranteed it with an oath,
The phrase "legal guarantee" is unique to the NWT (as far as a super quick search shows). However, the definition seems to be okay. The phrase used is is "eis bebaoisin" which is can be a technical term for a legal guarantee in a transaction. It literally means "for confirmation". The idea expressed in verse 16 is not uncommon. In the U.S., there's such a thing as a "verbally binding agreement," which, when tested in the courts, if both parties state that they agreed to something verbally, it is considered binding, thus, a legal guarantee. Verse 17 here is showing the binding agreement God entered into when he made his promise to Abraham. It really isn't that far of a stretch to say that all the covenants, which means "promises", are "legal contracts" bound by verbal agreements, sometimes by only one party (God himself) and other times by two parties (e.g. the Law covenant where the Israelites agreed to it verbally (Ex 19:3-8).)
So, at first glance, it appears that while the use of the phrase "legal contract" to the degree used in the article is irritating, it does not appear to be incorrect.