In England we would say "I would like an ice-cream" an ice-lolly is completely different it is more frozen (flavoured) water and not ice-cream.
As to whether it's on a stick or not, that depends on what's available and what the alternative options are. For example, if I go into an ice-cream shop the options would be:
- A cone - which is an ice-cream held in a wafer cone
- A tub - which is an ice-cream held in a tub
- An ice-cream on a stick
For each I would say the following:
- I would like a cone
- I would like a tub with 1 (2, 3, 4...) scoops
The ice-cream on a stick is something pre-manufactured and not made in the shop so I would always refer to it by it's brand e.g "I would like a magnum" or I would simply point to a picture of the ice-cream I want and say "I want that one"
On the other hand, if I'm in my house and my kids ask me for an ice-cream, that generally means whatever is in the freezer at the time. If I had multiple options of ice-cream there would always be a follow up question "Which do you want?" and they would either say a flavour which would typically imply free-form ice-cream unless I have multiple flavours of the same ice-cream on a stick where they would have to fall back to the brand of ice-cream.
In my 29 years living in England I've never heard anyone refer to it as an ice-cream on a stick.
This is a great question, I never realised something as simple as ice-cream could be considered confusing but when you think about it it really is!