The language you speak is English; the dialect is American English (or rather, American English is a group of dialects, one of which you speak). Similarly, British English is also a dialect of English, even though it can be thought of as the "original" dialect.
Dialects are defined precisely because languages vary in different regions, be they small or large. Just as I would say soda to refer to a carbonated beverage, others might say pop, coke, or one of many other terms. That doesn't mean we speak different languages; it only means that we use different words to refer to a specific concept.
At the heart of these dialects (or families of dialects), the core of the language is the same, even though the vocabulary may vary slightly. I can easily read a book in British English and don't have to "switch" my brain over to process the words differently—I may just have to remember the small variations in each language from time to time.
The question you may ask, however, is what separates a dialect from a language and when the former becomes the latter. To that, there is no definitive answer—only what is commonly accepted and understood both by the speakers of the dialect/language in question and by linguists who have studied the history of how they have evolved.
Wikipedia offers some influencing factors, especially noting the politics of the regions. This is true, but you should always keep this in mind: "How hard is it to understand a speaker of Dialect/Language B?" Etymological factors definitely make understanding related languages easier, but note the differences between Spanish and French (both Romance languages)—and then look at American and British English.
Words like 'the', 'a', and 'of' are often called syncategorematic words, words "that do not stand by themselves... (i.e. prepositions, logical connectives, etc.)" (here).
Examples of syncategorematic terms include:
- articles (for example, 'the' and 'a')
- connectives (for example, 'and' and 'or')
- prepositions (for exmaple, 'in' and 'at')
- quantifiers (for example, 'some' and 'all')
These contrast with categorematic words, "words that designate self-sufficient entities (i.e. nouns or adjectives)" (here).
Merriam-Webster defines categorematic as "capable of standing alone as the subject or predicate of a logical proposition : expressing a complete substantive meaning" (here).
Categorematic words include names (for example, 'John') and predicates (for example, 'tiger' and 'smokes').
You're correct in pointing out that syncategorematic words seem constant in some sense. That's why logicians and semanticists studying them call them logical constants. Their meanings do not vary from interpretation to interpretation (unlike names and predicates) (see here).
Bertrand Russell (1872-1970) was one of the first scholars to think about the meaning of the English definite article 'the'. He said that 'the' works like an existential quantifier, but with uniqueness. On his view, a sentence like 'the F is G' means:
- There is a unique F and it's G.
If you think about a sentence like 'The president of the US smokes', you can paraphrase it as 'There is a unique president of the US and he smokes'. Russell thought that this was the meaning of 'the': to express unique existence.
Similarly, he thought that the indefinite article 'a/an' had the same meaning as an existential quantifier. Thus, a sentence like 'A dog barked' means 'There is a dog and it barked'. The meaning of 'a/an' is to express existence.
Both of Russell's claims are quite controversial among logicians and semanticists concerned with natural language. They have spawned a century of literature. But that should get you thinking.
By the way, there is a semantic difference between your two "sentences":
- The play starts at 10:00pm at Rosedale
- Play starts 10:00pm Rosedale
As a matter of semantics, only well-formed or grammatical sentences express something. The second sentence is not well-formed and thus has no semantic content. Somebody who heard this "sentence" would be able to pragmatically interpret it, but this would likely involve their restoring the omitted 'the' and 'at', i.e. converting it to the first sentence. If you're wondering why English makes use of articles and prepositions when other languages get by without them, well, that's a different question entirely.
Best Answer
A lexeme is a lemma (what you called a “'base' word”) plus its inflected forms. In linguistic articles, you often find lexemes displayed as the lemma in small capital letters.
It's also useful to say what a lexeme is not: