I want to add to some of the excellent answers already given and don't have space to say everything within a Comment.
The term "word" in English seems to have a clear meaning, but actually just means a unit of language that, when written, is usually surrounded by spaces or punctuation and is without internal spaces. It doesn't have a clear meaning applied only to speech; nevertheless, because of how English grammar works, it is a convenient concept for English speakers to use. If you are talking about very colloquial speech with lots of contractions, the concept begins to break down, since the written form is arbitrary. For instance, can you easily tell how many "words" are in this utterance: "ya gonna see 'em?" It would be a common spoken equivalent of "Are you going to see them?" in some forms of American English.
Mandarin can be written in pinyin, which introduces the possibility of distinguishing written words; however, no one besides lexicographers actually knows all the rules about how to use spacing and many of the rules go contrary to what an English speaker might anticipate, based on English grammar. For instance, one dictionary I have even lists sentences with what seems to be inconsistent spacing, transcribing various groups of characters as "zuò xià le," "zuòxià le," "zuò xiàlai," and "zuòxià."
I think the word 单词 is mostly a technical concept that doesn't have an obvious use in daily speech. That is like the terms "verb phrase" or "main clause" in English that people may know from schooling, but don't use in daily conversation. The word "phrase" has a technical meaning in linguistics, but this is ignored in daily speech. I think that 单词 is similar and is not clearly defined or used for common speech.
When I look up 单词 in my copy of the 汉语大词典, I get in Chinese:
- 语法学用语。即词。与“词组”相对
("Grammatical terminology. Namely, a term. Comparable to a phrase.")
- 语法学用语。指单纯词，区别于合成词。
("Grammatical terminology. It indicates a single-morpheme word, which is different from a compound word.")
These definitions support the fact that 单词 is specialized terminology and further suggest that it is narrower in scope than what English speakers think of as "words," since it excludes "compound words" in one meaning.
I think Chinese speakers default to the use of 字 for talking about language, even when spoken. If there is a need to talk specifically about groups of words, I think the character 词 is used and defaults to the meaning "expression," with the number of characters being unspecified. I think that an expression like 常用词 does not clearly refer to "commonly used words" but rather to "commonly used terms."
From the Chinese perspective, it is unclear and unimportant whether a term like 下班 is one or two "words," whatever that means as applied to written or spoken Chinese. That is why a 词典 does not define words; it just lists various terms or expressions. It is not important or possible to define what are and what are not "words" when talking about sequences of characters.
Even in specifying the rules for writing pinyin as separate "words," the government seemed to have no use for the term 单词. Instead, the 2012 standards just refer to 词 (terms/expressions) and say things like:
拼写普通话基本上以词为书写单位
(followed by a list of one-syllable and two-syllable words and one three-syllable word, túshūguǎn (图书馆))
and
表示一个整体概念的双音节和三音节结构，连写。例如:
(followed by a list of two and three syllable "words," including quánguó (全国)，zǒulái (走来)，chángyòngcí (常用词)，and duìbuqǐ 对不起))
and
单音节词重叠，连写。双音节词重叠，分写。例如:
(followed by a list of expressions, including rénrén (人人)，hónghóng de (红红的)，yánjiū yánjiū (研究研究)，and shāngliang shāngliang (商量商量))
Rather than using the term 单词, which would be confusing to most people, they talk about whether expressions of various syllable length are written together or written separately. There is no need to refer to whether an expression like 睡觉 is one or two "words," only whether the parts should be written together or separately.
A definition of "word" in Chinese cannot use character writing as an arbiter, since all characters are written together, and cannot rely on pinyin because it necessarily uses arbitrary rules. The concept of "word" has uncertain application to the many Chinese morphemes that sometimes are independent and sometimes bound to others like the 识 of 知识 and 一字不识. The concept is also hard to apply to other morphemes that seem like suffixes, but may be separable in other grammatical contexts, like 看见 and 看得见.
In the end, pinyin is written in "words," but these do not share a clearly definable property beyond being separated by spaces and punctuation. In any case, these "words" correspond to both 单词 and 合成词. This marginal practice within Chinese speaking society is not enough to create the necessity for a common term to describe it beyond just saying something like 连写词 when talking about pinyin.
