If Apparation cannot be tracked, then why did it matter that The Department of Magical Transportation had been taken over?

Question

In chapter 11 of Deathly Hallows, Lupin claims: "It’s impossible to track anyone who Apparates, unless you grab hold of them as they disappear!". I'm not sure if this true. I was certain that one of the premises behind the plot to move Harry at the start of the book was that The Department of Magical Transportation had been taken over, and therefore Apparation could be tracked. What have I misunderstood?

I suspect that some quotes showing what taking over the department actually meant will be helpful in forming an answer.

Answer 1

Here is the explanation of the problems with trying to move Harry in the beginning of the book:

Pius Thicknesse has gone over, which gives us a big problem. He’s made it an imprisonable offense to connect this house to the Floo Network, place a Portkey here, or Apparate in or out. All done in the name of your protection, to prevent You-Know-Who getting in at you. Absolutely pointless, seeing as your mother’s charm does that already. What he’s really done is to stop you getting out of here safely.

The fear was apparently not that someone would be able to track where they apparated to, but that someone would be arrested for criminal activity.

Answer 2

The Trace

Harry was underage, and thus magical activity in his vicinity was trackable by the Ministry (although the Trace could not ascertain the identity of the caster, hence the false accusation about the Hover charm in CoS, which was actually cast by Dobby). Presumably, this would have included side-along apparition.

Once Harry had come of age (17 years), he was no longer subject to the Trace, and thus could apparate (whether individually or side-along) without said apparition being trackable.