Yes. The three categories there are not really reflective of many aviation businesses around the globe.
Formerly full service, not yet LCC
In many parts of the world (especially Europe) you have older carriers desperate to reposition themselves as low cost carriers. They have inherited large cost bases and they are trying to shed them, often with limited success. They often adopt the tactics of a "low cost" airline such as charging for amenities and operating on point-to-point routes. But their cost base (per available seat mile) remains as high as even the legacy, global carriers.
So they look like an LCC but actually they would not be considered an LCC by an investor. (Indeed many of them remain owned by sovereign funds.)
In the UK, Flybe (BE) might be a good example, although it is not in government ownership. BE charges low(ish) fares, it focuses on point-to-point routes, most of its routes are under 500 miles, it charges for snacks, it charges for luggage and there is no business class. However, until very recently it had a typical cost per available seat kilometre exceeding that of British Airways shorthaul!
LCCs are Flybe's biggest competitors and their lower average unit cost across their networks allows them to under-price Flybe when they do compete on the same city pairs. Even without direct competition, LCCs have a downward impact on pricing in many markets as they compete for discretionary travel spend. This could become a major threat to Flybe, especially if the level of overlap with LCCs grows.
On the flip side BE offers feed to airlines such as KLM, Cathay Pacific and British Airways. It also wetleased aircraft to Finnair in a joint venture with them called Flybe Nordic (an affiliate member of oneworld). These are not activities typical of a low cost carrier.
I think I would also put CityJet (WX) in this category. It was recently demerged from the AF/KLM group. WX does have a "base" at London City Airport (LCY), but it really focuses on point-to-point rather than transfer traffic (LCY is too expensive to compete for transit traffic). Costs and fares are high, but it's a good airline.
This weak unit revenue trend, at an airport [London City] that is considered to be high yield, is troubling, especially given the capacity cuts. Although the relative trend of RASK [revenue per available seat-kilometre] versus CASK [cost per available seat-kilometre] started to move in the right direction in 2013, it will be a considerable achievement to restore breakeven and then to convert that into sustained profitability.
LCCs now offering full service amenities ("hybrid" carriers)
Many of the original low cost carriers are close to saturating the "low cost" market in Europe and realize the only way to continue growing is to offer fares and service levels that are acceptable to business passengers—or indeed, discerning leisure passengers.
EasyJet (U2) has a new CEO who is making a strong push for business traffic by using central airports, better times, reliability, offering fast lane access, a loyalty programme called "EasyJet plus", and so on. Revenues have grown strongly.
A direct competitor to Flybe on UK domestic routes, whom I use quite often, is a relatively new airline called Eastern Airways (T3). T3 goes the opposite route: luggage is free, all of its passengers get fast track at all airports, it provides a free lounge at its hub, there is free coffee/beer/wine/snacks on board. Yet it charges low fares (most of the time), offers no feed or interline connections, does point-to-point only, and it only operates a fleet of turboprops (and one SAAB).
(However most of their planes have fewer than fifty seats, so maybe this example isn't directly relevant.)
“We’re offering a frequent business schedule with an all-inclusive on board service, both of which are equally important to business travellers.”
All premium long haul carriers
Other posters have mentioned the annoyingly named La Compagnie (try asking "Pardon, où est l'enregistrement pour la Compagnie?" in CDG...) offering all-busines service on PAR-NYC and LON-NYC. There have been one or two of these airlines in the past but they always struggle without the benefit of a major airline to rescue their passengers when the plane goes technical. Odyssey Airlines wants to try the same thing on LCY-JFK, but presently no aircraft can do the route without a stop.
However all of these operations are relatively "niche" and are only operating with about 30-50 seats.
Low cost long haul
Some think this will be the future. The obvious examples are Norwegian (DY) which operates a number of all economy services between Europe and USA and a few between Europe and the far east. I think they deserve to be in a separate group to low cost short haul. The challenges of operating a long haul service are quite different to low cost short haul; on many routes offering only economy and economy plus products is going to be difficult to sustain without sophisticated revenue management work to milk every cent out of those passengers.
Eventually, these operations will require feed from other low cost carriers, they'll require local "hubs" or interchange points, and that's when things will get interesting.
AirAsia X is one example of an airline already going down that route. It has a strong relationship with the various AirAsia brands and offers full interlining between shorthaul and longhaul services. Indeed, the low cost terminal at KUL had to be redesigned to enable bag-collect and re-drop prior to customs/passport control to enable better flow.
I would agree that many of the airlines I've cited here to rebut your three categories are a little bit exceptional, but I think AirAsia X and EasyJet are the airlines to watch because they are not constrained by starting in one category. The three "niches" you gave will not be hard dividing lines for long.