Gio:
UNDERTAKE has strong implications. In the simplest connotation, it just means "to begin" something.
But the stronger connotation is that it means to assume a powerful responsibility.
Debt of course, is a money issue. So if one "undertakes" debt, it has serious implications.
It puts one at risk.
For example, if I adopt a child, I UNDERTAKE a serious responsibility for that life.
In business, any project a company UNDERTAKES can involve millions and even billions of dollars.
So UNDERTAKE is any beginning with strong implications.
There are exceptions of course. I can UNDERTAKE the task of answering a student's question here on Italki. I can UNDERTAKE the making of a pot of coffee.
Generally though, the word UNDERTAKE implies that some serious matter of personal responsibility is the issue.