The bolded part is what is now called "quiet preboard". There are no gate announcements for this preboard group (hence the quiet reference). Basically, the computer system will let you know who is in a wheelchair, requires meet-and-assist, infant, etc. The GA goes from this list to determine the preboard group. Once quiet boarding is complete, then general preboard and the announcement starts.

AC appears to have tossed a bone to children/families section and allowed them to board early rather than be restricted to row number.

Note one can still get into the quiet board sequence if they have extenuating circumstances. Therefore a family travelling with a 2 yr old in car seat can get onto quiet boarding.