The Form 1099-R is almost certainly correct. Code H is only used for distributions from a designated Roth account in a 401(k), 403(b), 457(b) or the federal TSP, and it's implausible that such a plan would mistakenly use code J. The code J indicates that the distribution was from none of these but was instead a distribution from a Roth IRA.
Your implication is that you deposited these funds into another Roth IRA. Enter the code-J Form 1099-R as received, indicate that you moved the money to another retirement account, then indicate the amount that you deposited into another Roth IRA within 60 days.
The fact that you received this Form 1099-R implies that the movement of the funds was not done by trustee-to-trustee transfer but that the funds were instead paid to you and you made the deposit into another Roth IRA. If the movement of the funds was actually done by trustee-to-trustee transfer, the transfer is not reportable and the original Roth IRA custodian should not have issued a Form 1099-R at all. in that case you would need to obtain a corrected code-J Form 1099-R from the Roth IRA custodian showing $0 in box 1.
