I don't have an answer for you. I have more questions.
But I think you're asking good questions. By your question, you're an experienced nanny.
What are the parents like? Are they very anxious?
First comment above is a good one. Exactly what circumstances cause what reaction?
What do the parents do to prepare the child for you coming on duty?
Do they use you ...
should I be approaching it differently?
You say you sulk for a little while. I strongly suggest you don't. Simply disregard what your son says about not liking/loving you. Most likely, it's not true. Instead, immediately answer him with something like "well, you don't have to like me. I love you very much anyway, and always will."
I don't think ...
I have an anecdote which might not really prove all that much but might help alleviate at least some peoples worries. I'm happily married live with my wife and have a lovely 6 year old daughter. We (me and my daughter) have always had what I felt was a pretty good relationship though I am usually the stricter parent.
Around the age of 5 my daughter started ...
The things we hear about "love" as a child (and adult) are confusing, conflicting, and often do not at all reflect how "love" actually feels.
Radio plays songs that describe "love" as a most extreme feeling. Same for half of anything they can watch in movies or television.
(Some) religion require you to "love" both ...