What hotend designs exist as paired all-metal and PTFE-lined versions? I know that the E3D V6 hotend exists in both the all-metal V6 version, and the PTFE-lined Lite6 version with the same mount and form factor.  

What other hotend designs are available like this?
Hotend Designes in General
Lined
Makerbot Mk8 &amp; e3D Lite6
The Makerbot Mk8 seen on many Chinese machines and the E3D Lite6 are both lined hotends, which means the PTFE liner goes down into the heartbreak and butts against the nozzle. This style is very easy to print PLA with, but can't print hotter than about 240 °C.
Makerbot MK10
A variant of the lined hotend is the Makerbot MK10, where the PTFE goes a bit into the nozzle. These nozzles have a larger diameter, the melt zone is very short.
All Metal
There are mainly two designs of consumer all-metal hotend out there at the moment.
E3D v6
Stereotypical by now, the All Metal hotend typically means E3D v6. The liner ends at the end of the heartbreak, the heartbreak of the original is necked down to reduce heat creep. The design is flawed in that heat creep can't be mitigated completely. While this design can print much higher temperatures than lined hotends, the heat creep can create trouble with PLA.
E3D Hermea
The dreamchild of the v6 is the Hermea (formerly Hermes), which fights the problem of heat creep by drastically altering the coolend design: The heartbreak is even slimmer than the v6 and has very little area outside of the cooling block. This cooling block is fitted with a somewhat stronger fan. In theory this leads to much better cooling in the upper filament path and getting heat creep under control. However, due to the manufacturing stop induced by COVID-19, there are not too many Hermea out in the wild. Some early to mid development information could be gained from Joel and Thomas Sanladerer.
Among the still few reviews I have found is Thomas Sanladerer.
Slice Engineering Mosquito Hotend
Slice engineering went a different way to fight heatcreep in an unined all metal hotend: The heatsink is mode away from the filament path and serves as its own structure. As a downsize, each and every part is pretty much precision machined, making it comparably expensive: without heater, thermosensor, and extruder the bare item clocks in 145$. Claimed benefits are, that the thinner filament path (as it doesn't need to be structural) from a low heat-conducting material reduces heat creep and because we have an unlined filament path it can print at 450 °C, making it one o the highest-rated hotends.
Slice Engineering Copperhead
Currently, the same company is working on the Copperhead, in which they introduce a dual metal construction on necked heatbreak. The name Bi-Metal for this however would be a misnomer: Bimetal is a standing technical term for a strip of metals that, when heated, starts to tilt into one direction because of dissimilar heat expansion. The correct term, which they properly use on their website is Bimetallic, not Bimetal.
Because of the dissimilar thermal behavior, such a setup needs to be designed and machined very carefully, but could, help to dissipate the heat from the filament path and massively reduce heatcreep. The Copperhead as shown currently is offered for preorder with two mounting options, one of them the e3D style groove., which might allow drop-in replacement of a full hotend.
The heartbreak however alone could suffice to get some of the effects, and as the website shows, it would be available as a drop-in replacement part. However, pricing and effectivity remain still to be seen.
Paired?!
At the time of this writing in May 2020, the combination of e3D v6 and Lite6 using the same outer form factor is pretty much unique to them and their clones. The Trianglelab Dragon hotend (a review of it here) uses the e3D v6 Style connector on a Slice Enginering Mosquito Hotend design, making it pretty much paired to the two.
However the main functioning parts of the distinction is the heartbreak being either part of the cooling body and lined or a separate piece. The e3D v6 does use the same M6 threading that is also used in a Makerbot Mk8 and many Chinese clones on many machines [Some are M7, so beware!], and thus one can easily make a Frankenstein Hotend by combining the pre-mounted cooling body with a proper e3D v6 heartbreak and turn the lined into an All Metal hotend.