Stricktly speaking the term rope only applies to corage over .39 in (10 mm ) in diameter.
Anything less than this is referred to collectively as "small stuff' which may be in decending order be referred to as cord, spunyarn or twine.
Domestic small stuff might be called string or thread.
Any rope of a particualr function is called a line as in washing line, tow line guy line, mooring line. Source Geoffery Budworth.
So, a mooring line is still a piece of rope. but stricktly speaking no beach cat would have any rope on it because the cordage is all 10 mm or less.