"Indeterminate forms" are a vague concept and it is better to keep them "vague" rather than define them properly.

Limit evaluations are done on the basis of certain limit theorems which include the "algebra of limits" in particular. Theorems dealing with "algebra of limits" suffice to calculate limit of expressions which are composed of sub-expressions combined with $+, -, \times, /$ and the hope is that each sub-expression has a limit (perhaps calculated by expressing is as a combination of sub-sub-expressions) and then we use the algebra of limits to calculate limit of the expression by combining limits of sub-expressions via operations of $+, -, \times, /$.

These rules of "algebra of limits" however have two main limitations:

1) Limits of sub-expressions must exist (meaning they are finite, sorry I had to be explicit here to use the word "finite" as some textbooks treat limit $\infty$ also as "existing").

2) Rule dealing with division says that the limit of sub-expression in denominator should not be $0$.

"Indeterminate forms" were conceived to enumerate the cases where "algebra of limits" fails because of the above two limitations and for each of these cases certain other tactics / methodologies were developed. A classic case is expression of type $f(x)/g(x)$ where both $f(x), g(x)$ tend to $0$. Now to classify such cases the indeterminate form $0/0$ was invented. Similarly to deal with expressions of type $f(x)g(x)$ where $f(x) \to \infty$ and $g(x) \to 0$ the form $\infty\times 0$ was used. Also in each case where an indeterminate form was invented all the following options were possible: 1) limit exists, limit is $\pm \infty$ or there is oscillation. So classifying certain cases into "forms" did not guarantee the eventual limit, it only allowed us to use tactics and tools suitable to that form. Hence the word "indeterminate" was also added (we could not determine the limit by the form).

Cases like $f(x)g(x)$ where $f(x) \to \infty$ and $g(x) \to 1$ can't be handled by "algebra of limits" but these are not classified into indeterminate forms because there are theorems in this case which say that the resulting limit is $\infty$ so that the form is no longer "indeterminate" and we could perhaps classify such cases into "determinate forms" if we wanted.

Coming to the form $1^{\infty}$ it is obvious that it is designed to handle expressions of type $\{f(x)\}^{g(x)}$ where $f(x) \to 1$ and $g(x) \to \infty$. In such case we can write the expression as $\exp\{g(x)\cdot\log f(x)\}$. By properties of $\log$ function if $f(x) \to 1$ then $\log f(x) \to 0$ and hence $\{g(x)\cdot\log f(x)\}$ is already an indeterminate form of the type $\infty\times 0$. Therefore $\exp\{g(x)\log f(x)\} = \{f(x)\}^{g(x)}$ also has to be considered as an indeterminate form and it is usually written in the notation $1^{\infty}$.