Generalised planets? There is somewhat of an abstract way of generalising the notion of planets. 

Standard definition of planets is, obviously: "planets are the objects formed from the residual material surrounding a newly formed main sequence star through a complex formation process". 

Let me introduce a bit more general, by no means commonly known, definition of "generalised planets as bound macroscopic objects formed from some former accretion process taking place near some gravitating body over many orbital periods". 

Clearly, the latter definition is more general, as it doesn't specify the type of accretion event and the type of the central body. Now, for example, the accretion mechanism can be very different from that in protoplanetary discs: consider, for example, accretion on a compact object (a neutron star or a black hole), accrection on supermassive black hole, accretion from remnants of binary neutron stars merger, or any other possible accretion process.

As noted in some answers, some stars also form in accretion-involving processes in the vicinity of other stars. To avoid including stars into 'generalised planet' category, the definition specifies that the objects should be formed over many orbital periods, implying that they should be sufficiently bound with respect to the primary.

The question: Are there generalised planets other than planets/exoplanets?
The main flaw with your definition of a "generalised planet" would be the boundary between stars and planets. And, in particular, the status of brown dwarfs, that are kind of "failed stars", that could not start to burn their hydrodgen.

With this regard, the definition of a planet as it is now is probably the best one (or the less worst one), because it is then possible to draw a line between planets and stars (and in particular brown dwarfs), because a planet is then an object formed by accretion in the debris material of a forming/newly formed star, whereas a star is, in general, an independent object formed by collapse and accretion (there are now observations of forming brown dwarfs; see for example the review by Luhman (2102) or the recent paper by André et al. (2012)). So you have two types of celestrial objects, well defined by two types of formation mechanisms (core accretion for the planets vs. gravo-turbulent collapse for the stars (brown dwarfs included). Then you can also clearly distinguish between "real" free-floating planets (that are then planets expelled from their orbit and floating around in the galaxy) and brown dwarfs (that form independently, where they lay).

Bottom line: if your object, in any environment (including any type of accretion disk) is form by core accretion: it is a planet. If your object is form by gravitational collapse (with a later, unavoidable phase of accretion onto the central object): it is a star.