What the first statement means is that the horses aren't resting when there isn't any racing in the break between game years. (Though actually, they kinda sorta rest at half-rate this pseudo-Week 1 when entries are being taken.)The rules say horses aren't rested during the break, but goes on to say that horses raced in Week 16, can race in Week 2, which by the first statement means that's a one week break between races, not the standard two.
Week 2 is two weeks after Week 16. So, if a horse races Monday of Week 16, and then Monday of Week 2, that's a two-week break.
Probably the simplest way to think of it is that Week 1 is like Week 17, and Week 2 is like Week 18. If you can keep that in mind, it hopefully won't mess up how you manage your horses.
When in doubt about how tired or rested your horse is, ask the vet!
If it makes you feel better, a lot of experienced players get confused about how the break between game years affects their horses.