Your undertsanding is incorrect.

covid19.nj.gov:
Travelers and residents returning from any U.S. state or territory beyond the immediate region (New York, Connecticut, Pennsylvania, and Delaware) SHOULD self-quarantine at their home, hotel, or other temporary lodging following recommendations from the CDC:
If travel is unavoidable, travelers SHOULD consider getting tested with a viral test (not an antibody test) 1-3 days before the trip and again 3-5 days after the trip.
If travelers test positive, they SHOULD self-isolate for at least 10 days and should postpone travel during that time.
If travelers test negative, they SHOULD quarantine for a full 7 days after travel.
If testing is not available (or if the results are delayed), travelers SHOULD quarantine for 10 days after travel.

There is no MANDATORY quarantine.
State health department SUGGESTS that you SHOULD get tested before and after travel.
It is YOUR decision wether to get tested, to quarantine or not to quarantine.

"The self-quarantine is voluntary, but compliance is expected. Travelers arriving from areas with increasing COVID-19 cases may wish to postpone their travel to the region if they are unwilling or unable to follow the quarantine advisory."

There is NOTHING that states children CANNOT participate in training or games.

Subtly incorrect. While there is no mandatory governmental requirement to quarantine, your kids soccer club/team could do so. We've received many communications from our club saying you must quarantine after traveling out of state and the same thing with the school district. No one can force you in those circumstances as this isn't a police state, but there could be consequences for not following the rules. I'm not going to jeopardize the health of others or my kid's position in a club but ignoring a clear request. Others will feel differently, but a business can have it's own set of rules to participate.

“Subtly incorrect” lol it comes from the state health dept.

So then confirming - from the NJ state web:
There is NO mandatory quarantine.
Children CAN train and play upon returning.
A club can have its own policy-that’s wonderful.
It is YOUR call if you CHOOSE to quarantine at home.

-Yes, incorrect.
-You can choose to do whatever you want as that poster said.
-It's not a state, federal, etc. requirement and we don't force people into quarantine like china - also as that poster said.
-Technically, you can go to florida and then come back and have your kid do whatever you want. That's YOUR right.
-It's also the CLUB's right to kick you out for violating one of their requirements for playing/training.
-Do whatever you want to do, but understand what your club is mandating before you do it.

What kids have a week off in February? I have kids in 3 different schools in Monmouth County due to different ages and none of them have a week off. The only school day off for all of February is Monday, February 15th.

We get it-you think people should quarantine. Clearly you didn’t expect a poster to quote from the NJ health department website clearly saying that there is no mandatory quarantine. Thanks to the poster that quoted NJ policy...That’s what the OP asked to verify.

Have fun at your tournaments, vacays, whatever 👋🏼

In Covid-life three weeks is a lifetime of possible changes. If the rules at the time are you should quarantine upon return or get tested, then do the right thing and do that.

I'm the poster you're trying to quote. It doesn't matter if I think people should quarantine or not, and I very clearly did see that the initial reply did quote the health department as I included as such in my reply. It's not a big secret. In fact, people have been throwing around that quarantining isn't a "requirement" since the beginning of the crisis. It comes up in every angry parent complaint in my kid's school district.

Actually, it's unclear what the OP is asking. They didn't ask about a specific requirement (state or private), just what should be done. If they only want to know about the state, then the answer is here. If they were looking for a little deeper of a response, then I gave it. The person is going to do whatever they want, but did ask for advice.

just because your team does not practice or play you cant call it a quarantine. just because your school will not allow you to go into school - you also cant call it a quarantine.

they are not telling you to quarantine, they are quarantining themselves from you.

if you go out and jump on a train, go to a store, get out on the field and play with friends.

that would be a mandatory quarantine.