My copy of Pearson"s Organic dearteassociazione.org (7e), Morrison and Boyd, under the section "Reaction intermediates", accords a succinct description on the structure of carbocations:

The central $C$-atom (of carbocations) is in an $mathrmsp^2$ hybridized state, for which the carbocations have planar geometry. The $mathrmp_z$-AO (atomic orbital) remains empty.

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The stuff in parentheses was added by me

Aided by this description, I conjured up the following "general" structure of carbocations:

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Though I pulled out the above image from Google Images, it was pretty much the same structure I"ve been visualizing this whole time...drawing my own would be messy

And as you can see, I"ve equated the "planar structure" mentioned in the book to "trigonal planar structure" (with an axial vacant $p$ orbital). This image of a carbocation"s structure in mind proved rather handy, and didn"t seem to be incorrect at all.

Wikipedia, on the other hand, doesn"t sound so confident about the central $C$-atom"s $mathrmsp^2$ hybridized state.

One could reasonably assume a carbocation to have $mathrmsp^3$ hybridization with an empty $mathrmsp^3$ orbital giving positive charge. However, the reactivity of a carbocation more closely resembles $mathrmsp^2$ hybridization with a trigonal planar molecular geometry.

(Emphasis, mine)

As you can see, Wikipedia doesn"t appear to (completely) endorse the $mathrmsp^2$ structure of the central $C$-atom.

I continued to keep the "trigonal planar" structure of carbocations in mind while studying them. This posed no hindrance until, I came across these carbocations (in a book not really worth mentioning):

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Created using PubChem Sketcher V2.4

I"ve faced multiple problems while trying to ascertain the hybridization cum geometry/structure of the central, positive $C$-atoms in those carbocations. I shall list them separately,

1) Issue with the Aryl carbocation

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I visualized this as a particular Kekule structure of benzene having lost one hydrogen anion, thereby leaving a positively charged carbon atom in the ring. Considering the bonds involving the positive $C$-atom (in the particular Kekule structure I put up), I see two $σ$ bonds and one $π$ bond. Also, the $mathrmC=C^+-C$ bond angle appears to be $mathrm120^o$ (just like the normal benzene molecule. I honestly can"t figure out the hybridization or structure/geometry of the positive $C$-atom here. I guess I should factor in the "delocalization of the positive charge" across the ring, but that hasn"t borne fruit (for me).

2) Issue with the vinyl carbocation

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I visualized this as an ethene molecule, having lost one hydrogen anion, thereby leaving a positively charged carbon atom (seen on the right end in the image). Here again, I see two $σ$ bonds and one $π$ bond. From my knowledge of the VSEPR theory, I suppose the $mathrmC=C^+-H$ bond angle is $mathrm180^o$ (i.e- linear). But I can"t for the world figure out what the hybridization of the positive $C$-atom here, is. Heck, I"m not entirely sure if I predicted the geometry (linear) correctly in the first place...well, this case is alien to me.

3) Issue with the ethynyl carbocation

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I visualized this as an ethyne molecule, having lost one hydrogen anion, thereby leaving a positively charge carbon atom (seen on the right end). Considering the bonds involving the positive $C$-atom, I see one $σ$ bond and two $π$ bonds. Hybridization? No clue. Geometry about the positive $C$-atom? Um...kinda looks like a ball at the end of a stick...not sure if there"s any "angle" existing ._.

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Could someone please address these "issues" I"ve encountered for the above mentioned (aryl, vinyl, ethynyl) carbocations? I"m not sure if assuming "planar" structure" necessarily means "trigonal planar structure"... or if there"s something about "hybridization" that I"ve grossly overlooked.

My question(s), more explicitly put:

1) What is the hybridization state of the carbon atom carrying positive charges in the three examples I"ve used above? How is it determined?

2) What is the geometry/structure of the said hybridized carbon atoms? If that isn"t clear: I meant along the lines of "If it"s $mathrmsp^3$ it"s tetrahedral, if it"s $mathrmsp^2$ it is trigonal planar, if it"s $sp$ it"s linear"

I"m still in High-school, so I feel a bit overwhelmed at the moment (trying to wrap my head around this...hopelessly)