Okay. So we have: ついたち, いちじつ, and いちにち. [ひとひ and ひとえ are both indicated as being obsolete]
ついたち is pretty well shown to be used for the 'first day of the month', so this usage is unambiguous.
This leaves us with いちじつ and いちにち. Which one is correct? いちにち is the more common form. The vast amount of the time this will be the reading. いちじつ is often used in compound kanji sets (四文字熟語, etc.）and would necessitate the kanji element 「一」 rather than the Arabic numeral '1'. The usage of いちじつ is more literary/poetic.
While you said that dictionary.goo.ne.jp didn't provide readings, the link that you posted in the comments was actually informative as to their usages.
To answer your second question, as to why these sites do not choose a 'default' reading, I would say it is because all readings are equally valid and their selection depends on context. Also, a 'default' reading indication can possibly encourage people to select an inappropriate reading and discourage them from selecting the most appropriate reading. Possibly having a 'usage frequency indicator' might be helpful though. You might want to ask them why they don't have something like this.