To take your naming idea further, there's root position, first and second inversions. This much you know. However, taking it further, add just an extra C on top... but don't stop there, because there's a plethora of other ways to voice those three notes, especially on a piano, or using an orchestra. In fact, so many varieties of that C-E-G combination - C at the bottom of the piano, a G in the middle, an E,G,C at the top is just another...I can't work out how many different combinations of 4,5,6 or more notes, but it's going to be more than lots! The problem is/would be, that they would all need special names, so the operation would get unwieldly, to say the least!
So, sorry, no, there are no particular names for voicings such as you suggest, so if you need to transmit a particular one to others, it could be done by writing out the dots for them to read, showing them on whatever instrument, or merely showing what voicing you need.