Does the following count as "spelling"?

Li-Mei Lin. Lin as in forest, and Li-Mei as in beautiful, but with the characters reversed.

Chinese names (and the language in general) is made of characters. So, in order to explain it to people, you have to point out which characters the name has.

Does this count as spelling? If not, what's a more suitable word? "Explaining"?

I would not say lin as in forest, but the word-sign lin means forest, and li-mei means beautiful. This is explaining the meaning of the Chinese word-signs. And I would not use the vague term characters. Character can mean a lot of things. Chinese signs are signs for a whole word, so word-sign would be the clearest designation. Often these word signs are compound signs for single words which taken as a whole express a new meaning.

This is not transliteration, and I would not use the word spelling to describe this. Do you just want to tell people how it works, or are you trying to do some kind of translation?

Here is a general explanation: Chinese writing uses characters, instead of letters, to write names. People will tell you how to write their names by giving other common compounds that use the same characters. For example Lin as in forest ...

You can't just use a single verb to get this across, you need to use a phrase, such as "tell ... how to", "explain how to", "describe ... by", etc.

later note

This is also the way that the pronunciation of a particular character is described. An example of this is in the English translation of the novel Rose, Rose I love you, by Wang Chen-ho:

While introducing Dr. Yun as the next speaker, he wrote the surname on the blackboard, saying that contrary to what anyone might think, it was not pronounced hui, as in a dirty face, or hun, as in the word for low-down bastard, but yun, as in pregnant, with child. The congregation laughed at this little joke.

The word is likely transliteration, which is defined as "the conversion of a text from one script to another."