"Audax" bikes are a fairly recent development.
At the conception of AUK, rules required all bikes to have full mudguards. This rule has been relaxed to be 'at the organiser's discretion'.
Most Audax riders used Touring bikes. Some decided they would ride Audax rides close to the maximum permissable speed, so installed Salmon Profile guards on their lightweights. Not bikes in the sense of 'road racing' bikes, but bikes similar to road bikes with heavier and more robust rims and larger tyres. They needed more clearance than full blooded tight Road frames for the mudguards rule.
When I joined AUK, my number was 1100 and something. After a few years away, I rejoind and my number is 6300 and something.
Many clubs organise Randos now, but 15 years ago, AUK members and CTC districts organised Randos.
Bike builders saw this popularity as an opportunity to 'cash in'. Special bikes were designed with mudguards and low gearing. ie the 'Audax' bike. Perfectly usable but not a true Road bike and not a true Tourer.
To say 'every bike that can be ridden on the road is a 'Road' bike' is a unmissable blooper.
Back in the day, a 'Road' bike was a balls-out racing bike, a 'Tourer' was as it says; and all inbetween was a 'sports' bike with the suffix of 'tourer' or 'racer'.
The sports bike was aimed at those who didn't race or hadn't the lolly to spend on a Road bike. It was likely to be cheaply built and attracted a good return for the shop and manufacturer. It was a huge market. As has been said, they gained the reputation of being a 'pile of shoot' so the 'sports' prefix was dropped to maintain sales.
Nowadays, there are so many different types and styles that these terms have faded into history. Shops label 'Sports' bikes as 'Road' bikes to sell them to non racers who like the thrill of crouching over the drops on a 45 mph downhill.
Then there was MTBs. Off-roaders, pure and simple. A great ride up and quickly down a forest track, brushing treetrunks and jumping ravines.
Very low gears.
Some cyclists had both a sports bike and a MTB. Then they wished the two were combined, and thus was born the Hybrid. Hybrid? What a dull name. What else could it be called. Town and Country??
And then Lance Armstrong happened. Trek and Spesh didn't start up by building Road bikes, they caught the MTB boom in California.
With Lance's success, the low cost 'Racer' was in demand so Trek and Spesh ( and Shimano ) reacted, and that's where we are now. Roadracing faxsimile bikes that don't cost the earth and can be entered in a Cat4 or Cat3 with confidence.
Grumpy old gits like me see tham as Sports bikes that have improved enormously since the seventies. They have Road bike geometry and look like Road bikes from the pavement as they roll past. Some have tripple chainsets and a 28 tooth large sprocket, what would only have been seen on a Tourer.
So what do we call them?
If you go in a decent LBS and ask for a Road bike, he will lead you to the £4000 + balls out Race bikes.
"Entry level racer"? "Intermediate roadrace bike"?
Road bikes go on the road, MTBs go on mud (and the road outside Mac Ds). 
