Travis is trying to give you more than you need.

I can tell from your URL and your garden that you are in England. Almost certainly your donor didn't buy them from "really-obscure-bamboos-R-us", (otherwise they would tell you precisely what you have) and almost certainly didn't buy a bamboo that would die in an English garden. There are only about 10 genuses of bamboo commonly sold in Britain, indeed unless you go to a specialist there are barely 10 species commonly sold. So even if we don't get the precise species and variety, we will probably identify it well enough for your needs. But we do need a close-up of the culms to do that. In fact even without a close-up, I would punt a guess that at least one of your bamboos is Pseudosasa japonica, the most commonly grown bamboo in Britain.

I can tell you one thing straightaway, you almost certainly have running bamboo. Bamboos come into two basic kinds - clumpers and runners. There are only two genuses of true clumping bamboos commonly grown in Britain, Fargesia and Thamnocalamus, (and "commonly" is stretching it for the second named) and even from a distance it seems unlikely that yours are one of those. So you have runners. Admittedly most runners only run fairly slowly in England (so most English sellers describe all the bamboo that they sell as "clumping" even when it isn't). But nonetheless before long your bamboo will probably be under the fence, maybe not this year, but perhaps next year or the following, whether you put it hard against the fence or at the front of the bed. So unless nextdoor's garden is solid concrete, you should consider whether your neighbour will mind having bamboo growing in their garden. If not, you should think about installing a proper underground barrier to prevent it spreading that way, and that will be easiest to do now. You might also want to enclose it fully with a barrier, so it isn't popping up where the slate scree is.