Lucy seems to forget that 'American' English and 'British' English often vary greatly. (I was surprised when I found that an American correspondent didn't understand my use of words like 'fortnight' and 'alight').
Shaneystar2's link provides a comprehensive answer to the question The most important thing about that information is that it shows that 'pudding' can be used in many ways.
For example, if someone says '"What are we having for pudding?", the word 'pudding' is synonymous with 'dessert' but, as the Wikipedis page shows, there are various specific meanings to 'pudding'.
The Concise Oxford Dictionary offers two culinary definitions of 'pudding':
Definitiion 1: 'Soft or fairly firm food consisting of ingredients mixed with or enclosed in flour or other farinaceous or ceral substance and cooked by boiling, steaming or baking'
That definition encompasses a wide variety of dishes including Yorkshire pudding (which is normally a savoury accompaniment to a meat dish but which some Yorkshire people eat as a starter, served with gravy, and some Scots eat as a dessert, served with custard), plum pudding (which Christmas pudding is derived from) and milk puddings (which are semi-liquid dishes baced on rice, semolina or tapioca).
(The use of 'pudding' to simply mean 'dessert' is given as a derivative of Definition 1).