the cold came back

  1. I had a cold that cleared up but then came back.
  2. I had a cold that cleared up but then the cold came back.
    Are both correct sentences?
    Does #1 mean #2?
    If both are not correct, please rephrase it.
    Thanks.

My two cents:

#2 is odd; don’t use it.

1 is OK but you might add a bit of a phrase in the end.

I had a cold that cleared up b but came back/recurred within a week.

Let’s wait and see what a native speaker has to say about it.

Tom

The only change I’d suggest to Tom’s example is to use ‘quickly’, not ‘soon’.:
I had a cold that cleared up (quickly) but…