"The retirement specialist that I spoke to told me that since 2 of the 3 Prudential IRA distributions were taken on the same day, the IRS will consider those as one distribution as they aggregate all my IRAs into a single IRA."
It seems that is not uncommon to get bad information like this from Vanguard. Having taking the distributions on the same day (or nearby days) is irrelevant. If the three distributions came from three different IRA accounts as you have described, that's three separate distributions, only one of which is eligible for rollover and under no circumstances are any combination of these permitted to be treated as a single distribution. As I stated before, two of the deposits originally classified as rollovers must be reclassified as regular contributions and removed from the Vanguard IRA by explicit returns of contributions.
With an explicit return of contribution (code 8 included in box 7 of the Form 1099-R), only any attributable investment gains required to accompany the returned contribution. With the recent marked downturn, it's unlikely that there will be any gains unless the funds are presently in a fixed-income account, in which case the gains will be small.
Because the rollover or conversion of the two distributions ineligible for rollover has not yet been accomplished, you have 60 days following the dates of receipt of those two distributions to complete indirect Roth conversions of those two distributions (regardless of what you do with regard to your Vanguard traditional IRA).
