DNA Replication Question-Lagging vs. Leading?

I know in the leading strand it takes a shorter amount of time because it's moving towards the replication fork I believe? And it's continually replicated with the DNA polymerase III adding the nucleotides to the strand, right? The lagging strand however can't move that way because it goes from 5' to 3'. As a result, the DNA polymerase III must add the deoxyribonucleotides in Okazaki fragments. The exonuclease removes the RNA primers after that DNA polymerase I replaces it with DNA. Is all of this information correct? I've been getting different takes on all of it. :/