Learn English – “A Day in the Life” vs “A Day in Life”

There's no logic for deciding whether to use "to start a day" versus "to start the day" in this case, just a personal choice. It's strictly a style preference that has no effect on the meaning in most cases: any day in general. This is different from the idiomatic American English "I went to the hospital", which is much more common than "I went to a hospital".

I'd use the hospital if I'd gone to the local hospital or to the one I always go to in my city (I have three choices here in my Taiwan hometown) and a hospital if I'd gone to a hospital in a different town or country. I might also use a hospital if my visit were for reasons other than a regularly scheduled medical appointment, a trip to refill my prescriptions, or a trip to the emergency room -- as in "I went to a hospital to buy one of those because I know they sell them in every hospital in Taiwan". But this logic doesn't hold for "to start {a / the} day".

The effect that it has on style in this sentence is minimal because both sentences are the same length, and the two words sound almost the same. OTOH, maybe I'd be more inclined to use the day if I were talking particularly about today rather than every day in general. Thinking a little more about the two phrases, I believe that I prefer the sound of the day, but I'm sure that it's a personal and idiosyncratic preference rather than a logical preference.

These two phrases mean two different things.

"An End, Once and For All" implies that this is an open-ended ending.

If this is from the Mass Effect 3 soundtrack as a simple Google search shows, it's probably pretty obvious to you that there were multiple endings and plot holes, ergo, the open-ended-ness of the song title.

"The End" implies nothing and lets us know quite firmly that this is the end.

It was a song that not marked the end of the medley, but also the last song recorded by the four Beatles collectively, and thus, appropriately enough, the end.

So it's important to note that context means (almost) everything, not only grammar.