As for initial letters in abbreviations, let's sort them below in more detail.
So, the diesel engine with an inscription of HDI is own development of PSA Peugeot Citroen carmaker on the basis of the Common Rail technology. Feature of operation of such engine that the producer regulates a considerable procedural service maintenance period. The owner of the car of these French brands equipped with the diesel engine can drive on HUNDRED for preliminary treatment once over 30 000 km of run. In addition, there is one more advantage. Replacement of a Serpentine belt and belt of hinged units isn't provided during all term of operation of the engine. By the way, the HDI motor not more loudly petrol even works at single turns. These new qualities allowed to borrow to cars with such diesel engine to 50% of the market of France.
Schild with TDI abbreviation for the first time I appeared on engines of the German concern Volkswagen. It designates the diesel motors developed with use of the principles of system of direct injection and a turbo-supercharging. The turbo-supercharging allows to receive the big power, bigger efficiency. Thus indicators of profitability remain at height as well as purity of an exhaust. TDI engines are rather reliable and unpretentious in operation. However, there is one shortcoming: rather small resource of the turbine (about 150 thousand km) because of high temperatures of a stream of the heated exhaust gases (to 1000 °C) getting on its krylchatka and high rotary speed (200 thousand revolutions per minute). But, thus, the resource of the engine can make to one million kilometers.
In operation atmospheric (naturally aspirated) diesel engines with direct injection are less expensive. Such engines disappear under identification mark of SDI. They are the simplest modern diesel motors which are designed on the Common Rail technology, but at the same time, possess the high level of reliability. To them big run aren't terrible.
As you could notice, the basic difference between diesel engines isn't present. All differences are explained by that each producer doesn't want to resemble another.