I will try to take a slightly different approach than the other answers, and examine in particular the standardisation issue.
It is a bit surprising that you wonder about this situation. The
Hopcroft-Ullman book (I use the 1979 edition) gives two definitions of
PushDown Automaton (PDA): acceptance by final state or by empty
If you read the section on Turing Machines (TM) construction
techniques (section 7.4 in 1979 edition), they explicitly state:
Designing Turing machines by writing out a complete set of states and
a next-move function is a noticeably unrewarding task. In order to
describe complicated Turing machine constructions, we need some
"higher-level" conceptual tools.
The rest of the section, and the following sections, show all sorts of
variations on the definition of a TM, extending or limiting the
definition, that are all equivalent.
The point is that, for each problem at hand, we will choose the one
definition that is best adapted too solving that problem. Of course,
each of these definition could do in principle. But depending on the
issue, some definitions will give you more perspicuity on a specific
problem, and thus make the proofs easier.
Considering the case of Context-Free (CF) languages and grammars,
there are many normal forms that have been defined: Chomsky Normal
Form, Greibach Normal Form, Binary form. All can generate any CF
language, and they could be seen as variations of CF grammar
definition. It is good to have them coexist, because each as a role to
play in some context. They are intertranslatable, but at cost.
If you try to analyze the cost/complexity of CF parsing they are not
equivalent, and this is to be taken into account. This complexity
issue is much more critical in the CF case than it is for TM, because
CF parsers are used a lot in engineering situations, while TM are a
purely theoretical tool. This does not prevent engineers using CF
parsing from using various grammar forms for the same language, in some
coordinated way, in order to take various issues into account.
In the case of CF languages, it can be a critical engineering issue,
and it was thus to be expected that CF gramars, and their different
forms would be normalized/standardized about as much as the size of wrenches or the
diameter of electric wires. Actually it went as far as defining a
precise syntax for writing CF grammars, the Backus Naur Form (BNF).
TM have few engineering applications that would justify normalization,
and they have much greater potential variability (which is however
partially taken into account by considering majors variations such as
multi-tape, multi-head, ...). This explain that there was little
pressure to adopt a standard form, given that mathematicians who use
them are supposed to be technically mature enough to be careful when
it can make a difference, such a counting precisely the number of
moves. Even for complexity, the small variations in definitions often
do not matter, because we consider only asymptotic complexity, and
often asymptotic complexity up to a polynomial function.
It is quite common in mathematics (among other sciences) that
different authors will choose different definitions, that are know to
be equivalent (or equivalent in most context), depending on their
taste, their view of applications, their vision of the problem
structure, their pedagogical purposes, etc. Definitions also evolve
with time as more gets known about a problem, and as perspectives
change. The same goes for notations. This variability is an important
source of progress and understanding. The Greeks were doing excellent
mathematics, but it is a lot easier with modern concepts
(e.g. variables), definitions and notations.
Standards are often seen as extremely convenient (cf wrenches and wire
diameters). But they can also a factor of rigidity that prevent
progress. Standardization is a double edged sword, so we must blunt it
a little bit for safety. The various definitions are usually not quite
identical, but close enough so that theories can develop more or less
in the same way.