How do you tell whether a certain word belongs to a certain word class/part of speech?
Well, simply put:
In grammar, a part of speech (also a word class, a lexical class, or a lexical category) is a linguistic category of words (or more precisely lexical items), which is generally defined by the syntactic or morphological behaviour of the lexical item in question.
If two words have identical
morphological behavior and identical
syntactic behavior, then the two words belong to the same word class. If two words have distinct
morphological behavior or distinct
syntactic behavior, then the two words belong to different word classes.
Once you have your words separated into different word classes, you can compare them cross-linguistically. If the words in Word Class A have similar morphosyntactic behavior to the words in the classes called "verbs"
in a lot of other languages, then it's probably useful to call Word Class A "verbs" in your language.
The question, then, is what morphological and syntactic behavior looks like.
Morphological behavior is usually pretty straight forward. This set of words inflects for number/case/etc., but that
set of words inflects for tense/aspect/mood, so this set of words is one word class and that set is another.
Where people trip up, I think, is that they consider only morphological behavior while ignoring syntactic
For example, a common spin you see is to let any given root take noun inflections or verb inflections or whatever other inflections—e.g., you can take a root meaning "rock" and conjugate it like a verb so it means "it was a rock." Then the conlanger claims that this means there is only one word class, because every root can take any morphological inflection.
But this ignores syntactic behavior. In the vast majority of systems that I've seen like this, the language still has syntactic verbs and syntactic nouns, and roots can only take verb inflections when they're being used as verbs, and they can only take noun inflections when they're being used as nouns. Such languages are cases of highly productive derivational systems, not single word classes.
The problem, I think, is that syntax is tricky and subtle
—it tends to be more invisible than morphology, so conlangers (especially beginners) aren't as conscious of it. Morphology has to do with the actual shape of the word, which is easily visible. Syntax, on the other hand, has to do with the relationships of words to each other. You can't just look at a word and see that—you have to see how it works in relation to other words.
One issue in syntax is how words fit into phrases. Can they act as the heads of independent clauses, or do they have to be arguments of overarching phrases? What kinds of phrases can they serve as arguments for? To oversimplify, words which can head independent clauses are often called the "verbs" class, while words which head phrases that serve as arguments for verb phrases are called the "nouns" class. So if, for a particular language, it's possible to say "When a word goes here/does this/looks like that, then it's the head of the independent clause, and when a word goes here/does this/looks like that, then it's an argument of the phrase," then you probably have a basis for a "verb" class and a "noun" class.
(For this reason, if it's possible to say that a language is SVO or SOV or VSO or whatever, then it almost certainly has a distinction between verbs and nouns.)
One approach that is not
sufficient for making claims about word classes (or most other claims about a language's morphosyntax) is this kind of "argument by translation":
Conlanger: "Here is my new conlang, it has no verbs".
Reviewer: "But what about that word there. It behaves in manner that looks suspiciously similar to how verbs typically behave."
Conlanger: "It's not really a verb. The sentence literally translates as 'Peter stabbing John', not as 'Peter stabs John'."
I don't much care how you "literally" translate any particular thing in your language into English—you must give evidence based on the actual behavior of the language itself
to show how it does what you say it does. Just translating a verb as a participle doesn't actually say anything besides "It isn't a verb because I say it's a participle, so there."
There's a reason
why we call things verbs, and other things nouns. Unfortunately, these reasons are rather subtle when you really get down to it. If you want to create a language with an unorthodox set of lexical categories, you have to read up on syntax, analysis, phrase structure, and related topics.