"During summer" vs. "during the summer"

Solution 1:

As someone commented, I work during the summer could mean a couple of things. It could mean that one works only during the summer. Perhaps it is a student who takes a job, and then explains to his friend:

Yeah, I don't have time to relax during summer break. I work during the summer.

However, this phrase may also take on the meaning that one works during the summer, as well as during winter, fall, and spring.

On the other hand I work during summer usually only implies that one is already working, and keeps working during the summer period. One example of this use might be a summer school teacher. He might say:

My students during the school year are great, but I work during summer to make ends meet.

Interestingly, during the summer is more common than during summer according to this NGram:

However, you could pretty naturally say either phrase in order to imply what you mean. In context, the listener will figure out whether you mean "summer only" or "including summer".

Solution 2:

I disagree with comments and answers saying that including the implies only summer, and not any other times of the year.

To a large extent it makes no difference, but I would say if there is a slight nuance involved, it's because the is the definite article. Which implies a specific summer, not summers in general.

Therefore OP's first version is slightly more suitable where you're only talking about the coming summer. The second version is more suitable if you expect to be working every summer.