So I checked the code and I was wrong, sorta. There is a difference between the culture distance and culture level on a plot as I did explain. But an in-depth readthrough of the code shows that this whole phenomenon is completely intended... but only under fixed borders. If you wish for RCS to behave in a more 'original' manner, select the No Fixed Borders option.
What happens is if you have fixed borders, the city will claim any tile that the old city had cultural influence over and range to claim as long as no other city also has range to claim that tile. Apparently later on if you don't have enough cultural level, you won't own the tile until you do, but this premature claiming feature makes the growth behave the way you are dissatisfied with, much because of what I did explain before about some of the shortcuts in the code that would be very difficult to try to program around (and this is NOT worth that sort of effort and ensuing delays.)
So, would you be able to live with turning off fixed borders? Truth be told I don't even see it doing much anymore since it lost its teeth a number of years ago during some turmoils over the subject.