What is the difference between currency manipulation, and a fixed exchange rate regime?
Correct, both a fixed exchange rate and a manipulated exchange rate involve the government setting this price. In BOTH cases it is usually implemented by promising to standing ready to buy or sell any amount at the fixed price. Of course, countries don't always have enough foreign cash to sell it at the promised price, nor do they always want to buy as much as the markets sell to them

The difference is, as you suggest, whether the trading partners (the US is usually the one that labels others manipulators) conclude that the exchange rate is fixed at a too depreciated level. This level keeps foreign goods cheap in the US and US goods expensive in the foreign country. 

Also, a fixed exchange rate is often set at a lvel that seems "too high", the only question being whether the government will manage to keep it so high, since it creates an incentive for its citizens to buy foreign goods. Instead a manipulated currency is deemed to be "too low".