I'm afraid different people may use these terms in slightly different ways. But the way I understand it,
- standard form 標準體 is one of several ways of writing a character, chosen as standard in a particular place and time (there are different standards: standard mainland forms, standard Taiwanese forms, standard Hong Kong forms, standard Kangxi forms, etc.),
- variant form 異體字 is a non-standard form of the character, that nevertheless is or was in use at some point,
- archaic/ancient/old form 古字 is a form that was used in the past, but is not anymore,
- simplified form 簡體字 (in the narrow sense) is a form that has undergone simplification in mainland China,
- inherited form 傳承字 is a form that hasn't undergone simplification, but is regarded as standard in mainland China,
- simplified form 簡體字 (in the wide sense) is a simplified form or an inherited form,
- traditional form 繁體字 (in the narrow sense) is a form of a simplified character before the simplification,
- traditional form 繁體字 (in the wide sense) is basically any form that has been used to write Chinese, other than the simplified form in the narrow sense,
- orthodox form 正字 is a form that was regarded as standard in the past, especially in the influential Kangxi dictionary (another term which refers to something similar is 舊字形),
- vulgar form 俗字, 俗體字 is a non-orthodox form of the character.
- I don't think alternative form is a well-defined term with a very specific meaning. In Wiktionary it seems to refer to different ways of writing the same characters in different languages (e.g. in Chinese and in Japanese), or to variant forms, when used in the context of a specific language. There are some Wiktionary editors here on Chinese SE, perhaps they can clarify this.
But as I said, these terms may vary. For example, 正字 sometimes means the same thing as 標準繁體字 (standard traditional form), and there are also contexts where it means 楷書 (regular script, one of the calligraphic styles).
And in case you are interested in the development of 明 and other ways it was written (such as 朙 and 眀), going all the way back to the earliest forms, such as , Chen Zhiqun wrote a whole chapter about 明 (chapter 5).
For a list of 古字, you can search for "old variant of", "archaic variant of", "ancient variant of" and "classical variant of" in the MDBG dictionary. I'm not sure whether “old”, “archaic”, “ancient” and “classical” are understood in different ways in MDBG.
For something more academic, you may want to take a look at the Dictionary of Chinese Character Variants, which contains over 100,000 variants, mostly ancient. But note that the dictionary is Chinese-only and has a terrible user interface.