I picked the term "garden hygiene" up from lectures on the history of gardening by David Tomlinson of Merlin's Hollow, Aurora, On. He was talking about the Victorian (formal) gardening in the UK. Because of the abysmal public health conditions at the beginning of the Victorian period, especially in central London, the Victorians mounted a huge and successful public health campaign (e.g. promoting the use of the new carbolic soap and crucially building modern sewer systems). David's point was that the Victorians extended this orientation to cleanliness to gardening. He also noted, however, that at that time gardening was a trade, in the same way that electrical and plumbing is now and notably that labour was cheap. So the large public and private estate gardens could employ teams of professional gardeners to maintain complex and beautifully manicured gardens.
In my usage, "plant hygiene" is about keeping the individual plants healthy. Exposing plants to sun, water, air flow and nutrients (all as required) is part of it, as is minimizing their exposure to pathogens. For instance, I'd guess healthy garden phlox plants can handle some exposure to powdery mildew spores (true some selections/cultivars more than others). I take my job to be to keep the growing conditions as optimal as possible.
Re the exposure to pathogens part: sometimes you see strong sentiments expressed on gardening sites, as to whether cutting back herbaceous perennials should be done before or after winter. I'd be the first to agree that how and where one gardens are relevant here; also that there's various things to consider. But I strongly believe that, at least with our closely packed and maintained mixed perennial beds and our climate, that fall cutting back results in a healthier environment for our varied perennials during their next growing season.
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