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ELU's answers aren't sufficiently specific, as decimals/(decimal) fractions (e.g. 222/100) can $\not\in (0, 1)$

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A fraction with absolute value less than $1$ (that is, less than $1$ and more than $-1$) is called a proper fraction.

So, the numbers you mentioned would be called positive proper fractions.

user_194421
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    What about $\sqrt2/2$? – Michael Hoppe Jan 18 '18 at 09:30
  • @MichaelHoppe The Wikipedia article (scroll up) states, "The word fraction is also used to describe mathematical expressions that are not rational numbers, for example algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2 (see square root of 2) and π/4 (see proof that π is irrational).". – user_194421 Jan 18 '18 at 09:39
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    Wouldn’t be quotient a better name? – Michael Hoppe Jan 18 '18 at 19:41
  • @MichaelHoppe It's not sufficiently specific. https://en.wikipedia.org/wiki/Quotient says that it's the result of division, without specifying that it is a result in $(0,1)$. – user_194421 Jan 18 '18 at 22:19
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In mathematics, the term "fraction" is wrong, as you said, because the fraction can be $\ge 1$ (or $\le 0$ for that matter).

In English language, in colloquial use, "fraction" is often meant to be a synonym with "a small part of", implicitly in $(0,1)$. Have a look at: http://www.dictionary.com/browse/fraction, any meaning apart from the scientific ones.

As the linked question is about English language, I would argue that the answer "fraction" is right, unless that word is used in mathematical context. If you say "Only a fraction of people I invited actually turned up", it means a small proportion, it cannot mean, for example, $\frac{100}{100}=100\%$ of people turned up. However, in a mathematical text this would be a gross imprecision, and, indeed, as the other answers say, you would need to say "positive proper fraction".