Yes - there is a distinction between a genomic intersection and a genomic element-of (subset) in bedops.
bedops --intersect is a literal coordinate intersection, while bedops --element-of is a subset feature. With the latter, you will receive the entire element if it meets your overlap criterion and all the extra columns are there.
Consider 2 files:
cat a.bed
chr1 10 100 id-1 2.34 SNV312
cat b.bed
chr1 5 77 file2 4.56 +
If I intersect those with bedops -i a.bed b.bed, I receive:
chr1 10 77
it isn't clear what extra columns should go in this intersected region. Should it be from a.bed or b.bed? What if a.bed has multiple elements in the intersection, or what if you use 100 input files instead of just these 2? Philosophically and generally, the intersected results make up a new BED file with regions that are different from elements found in either a.bed or b.bed as shown above.
If you instead use bedops -e 1 a.bed b.bed, you get:
chr1 10 100 id-1 2.34 SNV312
because the element in a.bed overlaps something in b.bed by 1 bp or more (you can change the overlap criterion from 1 to a different value or use a percentage, like 50%).
If you are looking for the literal coordinate intersection (--intersect/-i) and you want extra columns from 1 (or more) of your input files, then you couple bedops with bedmap. You first get the literal coordinate intersection with bedops, and then you map on information from the file(s) of interest. For example,
bedops -i a.bed b.bed | bedmap --echo --echo-map-id --delim "\t" - a.bed
will pull out the information in the 4th column of a.bed and map it onto the intersected regions. Other columns can be mapped too with --echo-map-score and --echo-map, which maps information from all columns. --echo-map is the most general, and you sometimes need a small awk or cut statement at the end to pull out particular columns of interest. We can help with that if needed.
Hope that helps.