Sometimes, it is pretty straightforward: if you work for any governmental institution, NIH, FDA, NASA, DOD, etc, under your J-1, you receive a salary provided by taxpayers from these institutions, thus governmental funds. Many government-sponsored study and exchange programs are usually short-term but also processed through J-1 and come with 2YHR and 2Y repeat participation bars, even though you sometimes spend only a few weeks or months in the US. It is usually less harmful as subjects of such programs are typically young. The program is short-term, so they stay in their home country long enough to fulfill the requirements. In other cases, it is less clear; for instance, if you worked in some private institution that governmental funds support, i.e. you were a post-doc working under the PI who received NIH or NASA funds, does that constitute governmental funding? You received nothing directly from the government, but your salary is still technically from the taxpayer's funds. I don't have a clear answer for it.
In some cases, DOS doesn't know itself. You may or may not be a subject, and your passport's DS forms or statement may not correctly reflect it. You need to file for an Advisory Opinion for them to decide in each case.
If you are curious, you can fulfill the 2Y-participation bar and not complete 2YHR. i.e., by living and working in a country that is not your home country. In this case, you can only get another J-1. The good news is that the J1 2YHR does not stack, so even if you get 2 or 3 J-1, you will not have to spend 4 or 6 years in your home country. The bad news is that all your accumulated time in your home country will be reset back to 24 months as soon as you enter again as the J-1. Oh, and by the way, O-1 trumps the 2YHR and participation bar but its requirements are seriously insane and it only allows you to work in the US indefinitely as soon as you retire, or somehow lose your O-1 sponsored job you need to leave.