Bob, if it flowers laterally instead of terminally, it is not an Aeonium (one exception). The one Aeonium which breaks this rule, A. simsii, flowers laterally and lives on afterwards. It is a distinctive plant with lots of little hairs along the edges of the leaves.
I am pretty sure that is not Aeonium lindleyi. There are two varieties of that species which I've seen, and I will describe them based on the Schulz Aeonium book. A. lindleyi var. lindleyi (Tenerife) is a very fat-leafed and pretty distinctive looking plant that grows sprawling stems. The leaves are often almost round in cross section. A. lindleyi var viscatum (Gomera) makes sprawling, open rosettes with very few leaves (for an Aeonium anyway). If you do a Google image search for this species you will see a large variety of plants, many of which are misidentified.
For what it's worth, that species is kind of special, because it is said to make an antidote to the Euphorbia "burn" (the sap of Euphorbias can cause extreme skin reactions). I can't speak for the truth of that, but I've heard it from more than one source.
In general I would advise extreme caution in identifying unfamiliar Aeoniums, to the extent that most of the plants in cultivation are hybrids or cultivars, and even the species that are present tend to have some hybrid character (like the clustering A. nobile that was going around a few years back). These plants change appearance dramatically with the seasons and the sun, during their annual cycle of growth and dormancy.
The Schulz book is outstanding because it strikes a good balance between habitat (mostly Canary Islands) and cultivation (and how they are different).
You can propagate your Aeonium from rosettes (short stems root faster and better than long stems) and you can always try to grow it from leaves (detach the base very carefully). They are often self fertile (I wouldn't know for sure about any of mine because there's always more than one in flower at any given time). The seed is very small and generally way more of a hassle than rooting cuttings, unless numbers or diversity are your goal.
One more thing.... the flowers of my containerized Aeoniums turn into catastrophic bug farms about 50% of the time, so I almost always terminate them in bud, for the good of the group. Plants in the ground benefit from a bit more predator activity and are not nearly as problematic.
