PROVIDENCE — The nursing-home lobby is fighting back against the state's "mischaracterization" of them as scofflaws refusing to repay $60.8 million in advances that were, in most cases, subsequently double-paid.The nursing homes say they are victims of the state's painfully slow process for determining who qualifies for Medicaid, which leaves them caring for people who need nursing home care for months without payment — in some cases, more than a year.