The specification in the IEEE802.3 spec is is based on cable performance, not designations. However it is commonly read that the below is true.
10BaseT requires Cat 3
100BaseTX requires Cat 5
1000BaseT requires Cat 5e
In the future please quote my entire thought if not my entire post, your partial quote makes it sound like I had stated something I hadn't.
Your statement that 1000BaseT will not work with less than Cat 6 is false. Amusingly 1000BaseTX does carry a strict recommendation of Cat 6.
I see stable 1000mbps connections all day long, I recommend Cat 5e or better because as you pointed out Cat 6 is expensive, especially with copper prices rising and the knowledge that performance mania won't be dissuaded.
I have personally run long term tests using a approximately 50m run between 2 gigabit switches using Cat 5 and have gotten (iSCSI data) throughput that was much better than 900Mbps. I don't know how familiar you are with speed tests in real world scenarios, however most people don't get that raw throughput with their hardware, let alone as data throughput after a complex protocol like iSCSI..
To be more specific my cable was run unshielded as part of a bundle of approximately 100 cables (plus being ran next to power and fluorescent lighting on the way) into a rack where our lab infrastructure resides. The interfaces were a DGE-550T and the built in interface on a DSN-3200, the switches were 2 DXS-3250s. This is an old 10/100 installation that we just dropped gigabit switches in on, not what I recommend, but not unusable.
That said I do recommend that anyone install the highest quality cable they can afford as it will future proof you, and provide the potential for better performance. The recommendations I provided are based on long held industry recommendations and just as I stated are not specified in any standards.
So, back to where we started, some things will work despite being below recommendation, there is nothing that tells either end what kind of cable is being used, if the run is short or your conditions are favorable you can get away with sub par media. I wouldn't hedge any bets on it though. With that in mind I will stick to my above recommendations.