Hmm while partially correct, it is not used in the sense in which the term was coined (depends on what precisely was said though, maybe a bit was misinterpreted).
Again: orthology refers to a separation by speciation. This means that it refers to two genes in two different species that derive from a single gene of the last common ancestor
For instance a species 1 (S1) splits into two species: S2 and S3.
S1 carries a gene A. S2 got a copy of this gene (gene A2) and S3 got a copy (gene A3). Gene A2 and A3 are orthologs of each other.
Paralogy is more complicated, it refers to separation by duplication, but it can happen either before, or after speciation. In other words, it can be found in a single genome, but it does not has to be.
Let us take the above scenario, gene A2 gets duplicated. We then got gene A2, gene B2 (its duplicate) and finally gene A3 (from species S3). Here gene A2 and B2 would be paralogs, whereas both would be an ortholog of gene A3.
But we can do more complicated. Let us assume that in the last common ancestor of species 2 and 3 there is a duplication event resulting in the paralogous genes X and y Now assume three species each carrying one of those copies, e.g. species 1 carries gene X, S2 gene Y and so on. despite the speciation, gene X and Y would be considered paralogs, as their split was due to a duplication event.
As you can imagine in very complicated trees it is hard if not impossible to really trace down the relationship between any two genes. Moreover, while it has been sometimes added to the definition, the function has in reality nothing to do with the classification as orthologs or paralogs. While in some cases orthology correlates with function, in many more other cases (transcription factors are notorious for that) it does not.
In any case, the original definition only tracks back their relationship, not their functions.