Both "all evening" and "all the evening" are correct, and both "all day" and "all the day" are correct.
Both "all" and "all the" + a time expression have the same meaning:
"all day/ all morning/ all week, etc.....through all of the day, the morning, the week, etc. -- use this especially when the day, morning, etc. has not finished yet: We've been traveling around all week / I haven't seen her all day – where is she?"*
However, as you can see in the examples from Google, below, "the" is often omitted (as the reference states) when the time period has not finished, or has just finished.
"¢ Today, I spent all day working on updates for the CEATEC website. As any of you who have ever ...
"¢ We've spent all day tidying. We've spent all day tidying out the garage here in the freezing cold and pouring rain ...
"¢ I have spent all evening changing the new layout I spent all day yesterday doing, which was changing the layout I spent all day ...**
In contrast, "the" is sometimes included to refer to a past time or a general time. In this case, "the" has a meaning similar to "that":
"¢ ... He was one of the strongest minions of Thardan and could not be defeated by weapons or magic words. I spent all the day chatting and gathering experiences.
"¢ He was always there at the start of every day and spent all the day slowly swimming around the pond greeting all the other Beanies, with a jovial Quack
"¢ And if you have spent all the day birdwatching, you probably wish to get some sleep yourself. To
In your sentences, Lucy may have spent all evening or all day (all this evening or day), or all the evening or all the day (all another evening or day) on the game.
While this omission or inclusion of "the" would be a careful distinction, the difference is not so frequently observed. Google lists 46,100 examples of "spent all day," and only 401 examples of "spent all the day."
In addition to the possible difference in meaning, the inclusion of "the" seems to give a slightly more formal tone.
*Longman Essential Activator. Addison Wesley Longman Ltd. 1997
**(Note: This sentence above –
"¢ I have spent all evening changing the new layout I spent all day yesterday doing, which was changing the layout I spent all day ...
could also easily appear as
"¢ I have spent all evening changing the new layout I spent all THE day yesterday doing, which was changing the layout I spent all THE day ...)