I don't see Rob for Robert, Ben for Benjamin, Becky/Becca for Rebecca as being nicknames. They are shortened versions of a name (and often quicker/easier to say).
My name is 4 letters which can't be shortened (eg "Fred") but can be lengthened ("Freddie"). I don't like the lengthened version of my name - it sounds horribly cutsie to me, but as it takes more effort to add the syllable, I take it as a compliment, assume the person saying it likes me, and never complaint.
I am occasionally asked "were you Christened "Frederick"?" to which I always respond "actually, I wasn't Christened, but it says "Fred" on my birth certificate".
I was always so jealous of people who had names you could do things with - the Elizabeths (Liz, Lizzie, Beth), Rebeccas (Bec, Becca, Becky, Rebeck), et al. I always swore I'd give any child of mine a choice.
We named her "Fred". I feel so guilty, but I love name.
I gave birth to her at about 4pm, so ended up staying overnight in hospital. Like most, not much sleep was had, so when we got home the next day, DH (who was desperate for it) had some 1:1 time with her and I went and had some sleep. When he woke me up to feed her, he said "her name is "Baby xxx"", so we called her "Baby xxx" for quite some time and now we call her "xxx". ONLY we do this, despite friends and family hearing us call her xxx. I love her name, and I love that we call her xxx as well. She's 14 now.