Yes we have ALL the kinds of 'fries'. 'chips' usually refers to what I think you mean by home fries and cottage fries. We have 'curly fries' and the kind from Burger King and McDonalds are 'fries', but the word is only used for the thinner kind. 'Chips' is an all-encompassing term for all these things - fries are a type of chip. I suppose potato wedges are the only kind that wouldn't be called 'chips'.
And what exactly ARE tater tots? I never got that.
What you call chips, we call 'crisps'.