I would say that an engagement is when you've decided to get married (he asks, she asks, or mutual discussion) and have or working towards a date.
Very long engagements used to be quite common, due to overseas work, military service, finishing education, parents wishes (particularly for young engagements), or waiting for the husband to become established enough in his career to support a family - but were still in the situation of getting married as soon as it was feasible.
'Engaged to be Engaged' has never made too much sense to me, as that just strikes me as saying that they're in a relationship that has the potential to be serious, but they're not sure yet. I also don't understand the case where a couple has mutually decided to get married, but isn't really engaged yet because there hasn't been a romantic proposal and a ring yet - they can be nice, but aren't required.
I do wonder sometimes about very long, open ended engagements when there's no active push to marry - say Sue and Joe are living together, have two kids and have been 'engaged' for the past five years, but show no desire to actually marry. I think in some cases that engagement is to pacify more conservative family members.