When a recipe asks (without clearly specifying) for 1 cup of sugar, should I assume powdered or tiny crystals form?
"Sugar" without any other description refers to granulated sugar. Granulated means crystals somewhat larger than the crystals in caster sugar, which is another commonly used crystal sugar in the UK. In other countries these smaller crystals may also be called superfine sugar or baking sugar. You can generally use granulated or caster/baking/superfine sugar interchangeably, but the volumetric measurement (spoons or cups) will be slightly different since granulated sugar is coarser than caster sugar. You might find this Sugar Conversion Tool helpful.
If the recipe wants you to use the powdered form of sugar, it will specify powdered, confectioners or icing sugar, which are all the same thing.
It asks you to use white crystal sugar.
If you needed powdered sugar, it would use one of the following words:
confectioner's sugar or
Some supermarkets sell "baking sugar" which is still in crystals, they are somewhat smaller, so they dissolve faster. Most recipes don't need this kind of thing, it is required to make your almost perfect cake one percent more fluffy, but an eater has to be pretty perfectionistic to notice the difference. Also, it only works with recipes which measure by weight. In almost all cases, you are better off to buy the plainest sugar you can find.