Heavy Cavalry -
1) Life Guards and Royal Horse Guards, often collectively known as Household Cavalry. Historically, they were descended from true "horse" regiments - ie units that wore helmets and body armour in previous centuries.
2) Dragoon Guards were the former line "horse" regiments, downgraded for economy reasons, but given the title "guards" as a compensation.
3) Dragoons. From the WSS period onwards and for the whole of the "Napoleonic" period, ie 1790-1815, British cavalry with the title "Dragoon" served as heavy cavalry only. I cannot think of any instances where they served dismounted, either partially or as an entire unit, and they invariably formed part of the army reserve.
Light Cavalry -
1) Light Dragoons were the original "light cavalry" of the British Army; formed in the mid-18th Century, they were initially the equivalent of a "light company" in an infantry battalion - not least because the dragoons themselves had given up their infantry roles. By the 1760s, these troops were so often extracted from the Dragoon regiments and brigaded together (much like grenadier and light companies of the infantry) that they were later reconstructed as individual regiments.
2) Hussars - by the early 1800s, contact with the various German and Austrian forces during the French Revolutionary Wars, led a few Light Dragoon regiments to adopt Hussar style dress and other characteristics (eg moustaches).
There were a few occasions where British light cavalry acted dismounted during expeditions overseas (examples include the Egyptian campaign of 1799-1801, the attacks on Buenos Aires in 1806-07, Maida 1806, and the West Indies). In such cases, this was purely because it was too expensive and difficult to transport horses such long distances, and in sufficient numbers. There was no intention to use these units as infantry for the duration of the campaign; the army was supposed to commandeer/capture horses from the local population or the enemy, and use them.
It is possible that the KGL cavalry, like most German armies, still undertook dismounted drills as part of their training, and all cavalry were supposed to be able to protect their horses and encampment, but it is worth noting that the British Army increasingly adopted a "mixed arms" approach to advance guards, pickets, reconnaissance and the like, and light cavalry invariably operated alongside light infantry (it was not unknown for the latter to ride double behind the troopers in emergencies).
One thing to note is that French dragoons (and those of major allies, such as the Italians) were the only such cavalry to continue carrying musket-type firearms throughout the Napoleonic Wars. All other cavalry types in virtually all other armies adopted the smaller carbines - often with very short barrels - which were suited only to very short-range skirmishing, usually from horseback. In my opinion, this is a big indicator of how other nations saw the role of cavalry.