What means by x86 , x64 , x86-32 , x86-64 and 32 bits , 64 bits in operation system server processor ?

What means by x86 , x64 , x86-32 , x86-64 and 32 bits , 64 bits in operation system server processor ?
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x86 := 32 bits
x64 := 64 bits
x86-64 := 32 bits but 64-bit capabilities

The term x86 refers to a family of instruction set architectures based on the Intel 8086 CPU. The 8086 was launched in 1978 as a fully 16-bit extension of Intel's 8-bit based 8080 microprocessor and also introduced segmentation to overcome the 16-bit addressing barrier of such designs.

The term x86 derived from the fact that early successors to the 8086 also had names ending in "86".

Many additions and extensions have been added to the x86 instruction set over the years, almost consistently with full backward compatibility.The architecture has been implemented in processors from Intel, Cyrix, AMD, VIA, and many others.

As the term became common after the introduction of the 80386, it usually implies binary compatibility with the 32-bit instruction set of the 80386. This may sometimes be emphasized as x86-32 to distinguish it either from the original 16-bit "x86-16" or from the 64-bit x86-64.
Although most x86 processors used in new personal computers and servers have 64-bit capabilities, to avoid compatibility problems with older computers or systems, the term x86-64 (or x64) is often used to denote 64-bit software, with the term x86 implying only 32-bit.