Part of the reason that local people don't often refer to street names when talking about those areas is that they're *very* obvious geographically. I've given people directions to the Commons by telling them to walk/drive downhill until they can't anymore.
I'm sure when you get here you will see it clearly, but I'll try to give you an idea of those places on maps.
First, some comments on your observations. You're right, there are very few whole houses or condos for rent. Most of the rentals are either broken-up large Victorians or purpose-built apartment complexes out near Lansing. Not much in between.
I wouldn't call the downtown area secluded by any means, but there are *very* few undergrads that live at the bottom of the hill. I live on a moderately busy street downtown and I still feel like it is very residential and quiet. I'll occasionally hear little kids playing in the yard next door or a barking dog, but never any party noise, and I feel quite safe walking around at night. It depends on where you're coming from, but it does not have a big-city feel *at all*. Going by street names, I would call the area bounded by Green St. to the south, Jay St. to the north, Lynn to the east, and Cayuga to the west to be safe bets, with plenty of other blocks outside of those borders being fine too, but it's a little bit more of a "see it in person" kind of thing.
Cayuga Heights also largely falls under the "see it in person" heading. It sounds very much like what you're looking for--very convenient to campus, also very secluded-feeling, a heavily wooded area. It is helpful to know where the frats are, though--and again, the students don't wander very far outside their sphere so if you're not literally next door to a frat it's very quiet. I'd say that anywhere farther from campus than the multi-street intersection on Highland Rd (Kline, Oak Hill, White Park, etc) would be fine--more than fine, very desirable. Triphammer Rd. past Hanshaw is the dividing line here (and a main route for many city buses), but houses in the neighborhood on the other side of Triphammer all the way over to Warren Rd. are also lovely. Less wooded, more suburban feeling. There are some apartments near 13 between Triphammer and Hanshaw which are fine starting-out types of apartments, very few undergrads and TONS of grad students, postdocs, and young professionals. Regular buses. Not very rural but very quiet; I lived there the first year I worked in town.
There are a number of apartments on Triphammer south of Hanshaw that seem to be popular with grad students not undergrads, though I haven't lived there personally so I can't say for sure. It's a very wooded area though, and feels quite secluded.
The densest student neighborhoods are in and around Collegetown, bounded by, hmm, let's see: Oak, Stewart Ave, 79, and 366. Those are the areas you'll probably be most strenuously avoiding. Though around 366 again tends towards more long-term residents, but it's a look-and-see area that can be super-quiet or not. Just past 366, again bounded by 79 to the south, there are a lot of desirable residential homes that I think you would find attractive--pretty, quiet neighborhoods, tree-lined streets, that sort of thing. Not a lot of bus service, but depending where on campus you are working, just a short walk.
East side of the lake, as you say, is very rural, though technically not far from campus. Mostly farmland over on that side, and they do seem to get more snow than anyone else. The west side of the lake along 96 is another area you might consider--there are several apartment complexes that serve the local hospital over there, somewhat regular buses, and lovely views. A few farms past the hospital.
South Hill--along 96B/Danby Rd--leads to Ithaca College. There are some student neighborhoods in that area, but mostly I'd argue against looking there because getting a bus from IC to CU is a hassle, more than because of student noise. I wouldn't recommend the denser-looking southern parts of Ithaca and the more rural areas to the south or west because of minimal bus service up the hill and because of a variable reputation. Same with any area within a block or two of Route 13 downtown.
I think that pretty thoroughly covers the town! Anything you're looking at that I missed?