Having just checked that website it refers to “teleconference at 12 p.m. EDT Monday, Oct. 26” ?
My understanding is that there is “12 o’clock midday” and “12 o’clock midnight”. Surely 12 p.m. would come 1 minute after 11:59 p.m. Similarly midday, coming 1 minute after 11:59 a.m. would either be 12 a.m. or even 0 p.m.
I believe that the NASA website is referring to their midday.