There is no single answer to this question and the answers depend on who you ask and for what purpose “word” is being defined. For example, in Japanese there are often transitive and intransitive pairs of verbs whereas it is normal in English that the same verb is used both transitively and intransitively. Those Japanese verb pairs are more likely to be considered as two separate words as the sounds are different (and the transformation isn’t regular so can’t be considered a simple inflection, although actually in many cases these pairs can be matched by historical inflections/auxiliaries being appended). In English the way the word sounds is identical so people are more likely to consider it to be the same “word”. How about irregularly conjugating English verbs (is, be, are)? Or how about linguistic features which are expressed by word order in English but by dependent auxiliaries in Japanese? How about when a word has multiple non-obviously connected meanings (run a business, run away, a computer runs)? Or when a noun is compounded in Japanese but has spaces in English? Or English phrasal verbs which only express the specific meaning when considered as a unit and not as parts (blow up, blow out)?
Essentially even within linguistics the way you count words will depend on what you’re counting them for. If you’re counting to measure native speaker vocab sizes then saying that “run” is the same in all usages and with any conjugation makes some sense as a native will surely be familiar with all conjugations and probably know all common usages. For a complete beginner learner it may make more sense to count “run” and “running” as separate units. For an advanced learner this makes less sense but it may be useful for some reason (comparative research etc).
As for folk definitions, most native English speakers would say anything with a space on either side of it is a word and then perhaps quibble about whether things like “isn’t” is a single word. It’s not too important as the definition has no application in those cases.
For Japanese people, they learn to divide sentences into units at school with the smallest units being 単語. Normally sentences are more naturally divided by 文節 than 単語 though. (私は リンゴを 食べる is a more natural divide than 私 は リンゴ を 食べる - you’ve probably noticed these pauses in spoken Japanese as well). Outside of school, compound words will be considered to be single words by most Japanese people, as will compounded verbs like 生み出す etc. As you can already see the Japanese words for words (単語、言葉、語彙) already have slightly more specific usages than the English word “word” so it’s a bit easier to divide things up in folk usage (still hard in linguistics though).
Another consideration is that pitch accent shows word boundaries in Japanese. For example, most loanwords from English that would be made up of multiple English “words” actually become compounded into a single Japanese “word” that will have a single accented point instead of multiple (it’d have multiple if it were separate words). For example チョコレートケーキ would normally have a single accent on ケ instead of the expected double accent (on on レ and one on ケ) which it would have it were two words. Adding the suffix 的 will always make the word flat (I think 協力 is already flat though) so the pronunciation/accent changes as though the compound is one word rather than two that are stuck together. I’m not an expert on pitch accent by any means and still have a relatively strong foreign accent when speaking so take this with a grain of salt and do your own research.
As a last note, here you can find a brief discussion of the methodology behind the way words are counted for the balanced corpus of contemporary written Japanese which is a great resource. They count words in two different ways for the sake of their research. As you’d expect, having more granular ways of counting using meaning/usage as a benchmark would take a lifetime of manual work so broadly automated and rough morphological analysis is far easier: 形態論情報 現代日本語書き言葉均衡コーパス（BCCWJ）
For the short units used by the BCCWJ these would be two different “words”. For the long units they’d be one “word”. The figures for a native speaker knowing 40-50k words would correspond closer to the short unit words.
Hope that helps a little even though it isn’t definitive by any means. For learners the main take away is to kind of forget about vocab counts past the beginner stage as they start to lose all meaning.
