The first representation isn't depicting the black hole; it's depicting its accretion disk, which is indeed plate-shaped. The accretion disk is around the black hole, and - if it's a rotating (Kerr) black hole - it's perpendicular to the black hole's axis of rotation. Accretion disks are disk-shaped, and so the representation is disk-shaped.
The other is a common representation of a black hole, but it's misleading. It's an extension of the "rubber sheet" analogy, which describes the bending of spacetime. A heavy object, placed on a semi-taut sheet, will curve it a little bit. This represents ordinary matter curving spacetime. A black hole is represented in that image as bending spacetime by an infinite amount. The issue is that this representation embeds the two-dimensional representation of spacetime in three dimensional space, making it appear as if the black hole has this extra depth. Black holes are not embedded in an extra spatial dimension.
Note also that the small object moving near the edge of the black hole is also represented by a tiny funnel. The creator seems to have tried to take the analogy to the extreme, portraying the object's funnel as finite, and the black hole's as either extremely large or infinite, in an attempt to explain that light cannot escape from a black hole. It's not too accurate, but it works to get the point across.
By the way, black holes do have size (the central point singularities of non-rotating black holes are infinitely tiny, while the ring singularities of rotating black holes are in the shapes of infinitely thin rings). You might want to read up on event horizons. The radius of the event horizon of a non-rotating black hole is a finite number, the Schwarzschild radius. It is calculated as
In this case, the event horizon is spherical. In rotating black holes, it may be in the shape of an oblate spheroid.