If a recipe doesn't specify the type of vinegar, is there a standard type implied?
If a recipe asks for "vinegar", a standard type is implied, namely white vinegar.
There certainly are more vinegars than just the white and the cider one (white wine or balsamic are also often used).

However, cultural differences can play a role. I've found on wikipedia that English people put malt vinegar on their fish 'n' chips. If you would see a recipe for fish 'n' chips that calls for "vinegar", my guess is that malt vinegar is meant. I'm sure there are other cultures which use other vinegars as standard. You can always ask some people in your environment. If they all agree, you can assume that's your standard vinegar.