Depending on what version of the Illumina technology reads come from, they may have barcodes of varying lengths, and on both reads of a pair, or only on one read of a pair.

Note though, that with Illumina technology, the barcodes are read in a separate run or runs from the R1 and R2 sequences.

However, in the cases where the DNA insert is shorter than the read length, you will read through into the first part of the adapter sequence, and eventually into the barcode sequence, so some reads may have an adapter sequence and maybe also a barcode sequence towards the 3' end of the read. This is what trimmomatic removes.

In the link you gave from Brown Uni, the sequencing primer hybridises to the green part of the sequence, and you start getting the sequence of the blue part of the line (sequencing-by-synthesis). If the blue part is shorter than the number of sequencing cycles, you read into the yellow part (the reverse adapter), and you get the sequence of this adapter on the 3' end of your read.

There generally aren't adapters, except at the 3' end if your fragment size was shorter than the read length. The first link you showed didn't include the location of the sequencing primers, which likely would have clarified this for you.

Check this link https://ngsc.med.upenn.edu/Experiments/Public_Data/Media/Public/FASTQ-Files.html Barcodes are part of the 3' adapter and used to separate reads from different libraries when pooled and sequenced. If you are sequencing inserts larger than your read size say 180-300bp you are not expected to see any adapter sequences or parts of them in your reads. But nothing is perfect in sequencing and sometimes you may end up with part your data containing few bases at the 3' end of your read containing the part of adapter sequences. When using trimmomatic use the adapter sequences specific to the type of library you used. If there are any adapter sequences at the end of the reads it will trim and keep the 5' part of the read.