When and why was impulse drive invented?

Impulse and Warp drives do different things. Your non-canon answer is a canon one. Throughout Next Generation Impulse is used to change direction and momentum, which is what is meant by delta-v.

Warp drive does not alter your velocity or momentum, it moves the space around the ship. So once you are out of warp you need some means of propulsion. That propulsion began as chemical rockets, which would be called thrusters in ST, presumably extended through ion drives and progressively iterated on until reaching the present day impulse drives that are used in the various series. I would call ion drives the first "impulse" drive. So that would mean impulse was invented in 1959.

Different cultures use different kinds of impulse systems. I believe the Enterprise (in all forms) uses a kind of fusion powered magnetic propulsion.

Impulse drive was invented some thousands of years ago by the first man who threw a stone. Impulse is the integral of a force over time and is equivalent to the momentum it confers to whatever it is applied to.
An automobile is an impulse vehicle.

Hence the impulse engine was invented long before space flight. In the case of space flight is is probably any kind of reaction engine, since that is the most natural way to get a force in space. Ram-jet engines have been considered for space ship, that would collect reaction mass by gathering atoms and ions with a large force field and get momentum by accelerating it with an internal energy source. As for ram-jets in airplanes, the faster you go and the more reactive mass you get.

Impulse engine means engine working with classical Newton-Eistein physics in normal space.

Zethram Cochrane may well have invented and tested the first human Warp Drive (with a little help from the crew of the Enterprise-D) but he certainly didn't invent the Impulse Drive. That honour goes to the Vulcans who then came to Earth as a result of seeing his successful test. As you can see from their smooth orbital insertion and landing, their ship clearly has both Warp and Impulse engines.

Although the Vulcans (at least according to the Trek: Enterprise canon) are very reluctant to share technology, it's pretty clear from the opening of Enterprise that impulse drive was in regular use within 15-20 years of the events of "First Contact" (see the USS Emmette below). This strongly suggests that they shared this technology with us, allowing humanity to replace the dirty and inefficient reaction drives they used before this.