Are sequencing reads written differently when inserts and indexes are sequenced seperately? I was going over different sequencing library protocols and noticed that the adapter sequence can vary between protocols. In some adapter sequences the index sits between the primer binding site and the insert. In this case during sequencing the index will be sequenced and then the insert that comes right after it. In other cases, the adapter sequence is built so that the index is outside the primer binding site, i.e. index - primer binding site - insert (see two leftmost parts of the attached figure). Here the insert is sequenced using the Read1 primer and afterwards the index is sequenced in a separate reaction using the 17 index primer.

In this case are the index reads just added on to the ends of the insert reads in the FastQ files? Is there a way to tell from the FastQ files where the index was positioned within the adapter?
The order of sequencing in a dual index flow cell is:

Read 1
Index 1 (I7)
Index 2 (I5)
Read 2

Between each step there is a switch of the primer used (Read1-primer -&gt; Index1-primer (I7) -&gt; Index2-primer (I5) -&gt; Read2-primer)
There are differences in how the I5 is dealt with between dual- and single-read flow cells as explained in the Illumina Indexed Sequencing Overview. In this document there is a lot more detailed explanation of the sequencing workflow that may be helpful to you.
But as explained by the other answers, none of this will matter in most cases. Even if you decide to output the Index sequences they will be exported as separate FASTQ files.