I have lived in England all my life and I agree with most of the answers from speakers of British English above. I never had any reason to analyse the circumstances in which Brits have traditionally used 'can' instead of 'tin' as an abbreviation for a metal 'tin can', but I feel certain linguistic norms has been missed. These are:
If something that is purely a potable liquid is stored in a 'tin can' that is designed for that purpose, then Brits will invariably refer to the container as a 'can', eg. a can of milk/beer/olive oil etc.
If a 'tin can' contains something that is solid and edible or is made up of a number of discrete solid edible objects, then Brits will invariably refer to the container as a 'tin', regardless of its shape, eg. a tin of bicuits/beans/tomatoes/salmon etc.
If a 'tin can' is designed for containing an inedible material, whether liquid or solid, then Brits will always refer to it as a 'can' if it is taller than it is wide, eg. a can of petrol/motor oil/paint. This continues to be true, even if this 'tin can' is later re-used to contain edible solids, despite not having been designed for that purpose. For example, a petrol can that is repurposed for storing biscuits is still refered to by Brits as a 'can'.
If a 'tin can' is designed for containing and an inedible material , whether liquid or solid, then Brits will always refer to it as a 'tin' if it is wider than it is tall, eg. a tin of polish/glue/ammunition. This continues to be true, even if this 'tin can' is later re-used to contain potable liquids, despite not having been designed for that purpose. For example, a shoe polish tin that is repurposed to contain drinking water, it is still referred to by Brits as a 'tin'.
The distinctions above refer to 'tin cans' that are made predominantly of metal, whether of tin, steel or aluminium.
As far as plastic counterparts are concerned, Brits will generally refer to a plastic version of something they could describe as a 'can', also as a 'can'. For example, 'can' of motor oil is still refered to as 'can', even if it is made entirely of plastic.
With two notable exceptions, a Brit would NEVER refer to any plastic container as a 'tin'. The two notable exceptions are the plastic counterparts of the traditional metal tins of biscuits and metal tins of sweets (candy). These are exceptionally referred to as 'tins', but every Brit is conscious he or she is uttering the word 'in quotes' ie. the word 'tin' is used as a conscious metaphor when denoting the plastic contemporary counterpart of a traditional metal buscuit tin or sweet tin.
To those of us who grew up in Britain, these distinctions are natural and unconscious norms, in the same way that a regional accent is natural and unconscious to natives of a particular region. The usage isn't logical because, like all natural language, it's the result of cultural evolution, not human design.