How many types of Bandpass filters are there for image processing?
That article discusses just 3 types of filters (honestly a 2D Butterworth filter is the first time I hear)  and is clearly lacking any generality. 1D analog and digital filter design is a mature subject. However at some points 2D filter design dramatically differs from a 1D filter design,  as reflected by the active research on the area.

Fundamentally image data is exceedingly sensitive to phase distortion and any operations (including filtering) must therefore preserve the waveform shape as much as possible. (i.e. they have minimum, if not zero, phase distortion.) 

Both 1D and 2D FIR filters are prime candidates for this required linear phase property. Their extension to 2D is straightforward.

A Butterworth filter is basicaly an IIR type and hence it has nonlinear phase response especially at the transition band edges. However, butterworth filters do have the smoothest passband characteristics among the basic filter types.

In contrast Chebyshev types I-II and elliptic filters are IIR filter types, with Equiripple characteristics in the passband and/or stopband. They have much sharper transitions than attainable via any FIR or Butterworth filter given the same order N.Elliptic filters are considered as the optimal IIR filter which will have smallest transition width for a given order N, at an accepted maximum passband and stopband deviation. However I have not seen any of these 2D IIR filter types to be preferred over a carefully optimized 2D FIR filter.

For 2D filter design, there are also other methods such as frequency sampling or 1D to 2D circular transforms. A particular class of such mappings can produce practically optimal 2D filters.

Finally a variation of Parks-McClellan minimax filters can also be adopted to 2D , however with more difficulties, since fundemantal theorems of 2D polynomials are more complex and less fruitful than the 1D case.