In DNA repair, how is it determined which strand contains the error?
There are many types of DNA repair. Let us limit to the following three different types to answer this condition.

1. UV damage 

Exposure of a cell to UV light can result in the covalent joining of two adjacent pyrimidines (usually thymines), producing a dimer. These thymine dimers prevent DNA polymerase from replicating the DNA strand beyond the site of dimer formation.

In such cases, UV-damage specific endonucleases cleave both the 5' and the 3' end of the lesion. Along, with the lesion some normal DNA is also lost as an oligonucleotide. Here modifications are being made only on one strand, the other is taken as a template (correct) and the DNA in the lost part is replaced.

The UV-damage may occur on both the strands but locally, one is affected and the other is the template (since you can't have thymine both sides)

2. Base Excision

Normal DNA bases can get deaminated spontaneously or on insult to an abnormal base (not on DNA).The abnormal base is excised by glycosylases and gap is later filled by considering the other strand the template.

3. Mismatch repair

What you are talking of in your example is not a case of deamination or UV damage but a mismatch repair. There can be errors in replication too where such a case could set in. There are specific enzymes for this repair, but discrimination is based on the degree of methylation. GATC sequences, which are found approximately once every thousand nucleotides, are methylated on the adenine residue. This methylation is not done immediately after synthesis, so the newly synthesized DNA is hemimethylated (that is, the parental strand is methylated but the daughter strand is not). The methylated parental strand is assumed to be correct, and it is the daughter strand that gets repaired.

The above said mechanism of discrimination is only in the gram negative bacteria. To quote Wikipedia, 


  However, in other prokaryotes and eukaryotes, the exact mechanism is not clear. It is suspected that, in eukaryotes, newly synthesized lagging-strand DNA transiently contains nicks (before being sealed by DNA ligase) and provides a signal that directs mismatch proofreading systems to the appropriate strand. This implies that these nicks must be present in the leading strand, and evidence for this has recently been found.


There are other cases where there is damage occuring to both strands that I'd suggest for further reading.

References


Lehninger's Principles of Biochemistry, Ed. 6 
Lippincott's Illustrated Reviews, Biochemistry, Ed. 5 
Wikipedia. DNA mismatchrepair