Normalization in IBM hexadecimal floating point
I'm going to start with this famous quote from James Wilkinson's 1970 Turing Award Lecture, Some Comments from a Numerical Analyst.


  In the early days of the computer revolution computer designers and
  numerical analysts worked closely together and indeed were often the
  same people. Now there is a regrettable tendency for numerical
  analysts to opt out of any responsibility for the design of the
  arithmetic facilities and a failure to influence the more basic
  features of software. It is often said that the use of computers for
  scientific work represents a small part of the market and numerical
  analysts have resigned themselves to accepting facilities "designed"
  for other purposes and making the best of them. I am not convinced
  that this in inevitable, and if there were sufficient unity in
  expressing their demands there is no reason why they could not be met.
  After all, one of the main virtues of an electronic computer from the
  point of view of the numerical analyst is its ability to "do
  arithmetic fast." Need the arithmetic be so bad!


In 1970, Fortran had only been cross-platform for four years. A lot of numerical analysis was being done on IBM hardware, and System/360 in particular, but every CPU vendor had its own floating point format, and most of them (as Wilkinson indicated) were not designed by those who demanded high-quality floating point arithmetic.

Today, the industry has standardised on IEEE-754. It's imperfect, because everything is, but it's reliable and can be relied upon.

Wikipedia's definition of "normalised number" (or "normalized number"; I'm going to type it as my dialect spells it) is the current modern definition. It's the one you learned in high school when discussing scientific notation. It's the one used by IEEE-754. It's the one that's in all the modern textbooks.

But it's not the only normal form for floating point that has existed, and I think that's the source of the confusion here.

Interestingly, it's not the only normal form in use today! There is at least one place in modern programming where a different normal form is still in use, because it pre-dates IEEE-754.

The C standard library has a function to extract the parts of a floating-point number's normal form, called frexp. But the normal form that it uses is that the mantissa is in the range $\left[ 0.5, 1 \right)$, not the modern $\left[ 1, 2 \right)$. This is a notorious "gotcha" for numeric analysts working in C or C++ today.