This detail matters very little for the story - the issue is if government people investigated a family on suspicion of terrorism because they looked for kitchenware on the internet, not how the agency these people worked for is called. If they work for Department of Education SWAT team [1] it were as bad as if they worked for FBI. The problem is not that they didn't have the right badge but that they invade privacy for no good reason.

[1] yes, DoE has SWAT teams: http://reason.com/blog/2011/06/08/dept-of-education-swat-tea...