There are two kinds of cord cutting rituals but the idea of cord cutting is the same - though, mainstream media has altered the meaning of it.
 
Cord Cutting is usually done to cut the cords between two individuals, but it doesn't usually cut the connection (that is a different one) this is usually done for cutting the toxic and negative things that are surrounding two people. 
 
This allows them to see things clearly - more of an energy cleansing. Sometimes one of the side effects is for two people to look for each other and if true love is there, they ended up strengthening the relationship. This is mainly for removing negative attachments.