Answering this question requires defining "language" and "dialect". It's easier to do this by using the non-technical word "variety".
Language: speech varieties are different languages if they are not mutually intelligible.
Dialect: speech varieties are dialects of a language if they are mutually intelligible but differ in systematic ways.
Some examples of dialects of English
Britain: Received Pronunciation (or RP), Yorkshire English, Estuary English, Scouse, Norfolk English, etc. Through various historical accidents RP came to be the most prestigious dialect.
The USA: General American, North-Central, Midland, Greater New York City, etc. General American is the variety with the fewest obvious regional features, and is used especially in the media. Rick Aschmann has an excellent and detailed introduction to North American English dialectology.
The Wikipedia article that says: "British English is the form of English used in the United Kingdom. It includes all English dialects used within the United Kingdom." is simply using the non-technical term "form" to refer to the grouping of all dialects of English found in the United Kingdom.
As for the Wikipedia article on Singapore English, I can't see where it says that "'Standard Singapore English' is not a dialect, but a form of English". It does say that Singapore Standard English is very similar to standard British English (which I assume refers to RP). Singapore Colloquial English (aka Singlish) on the other hand, while based largely on English (as well as on the other languages of Singapore) is so different from English that it is not usually considered a dialect of English (ie it is not mutually intelligible with any other dialect of English) but a creole.
So in short, referring to a "form" of a language is a non-technical usage that could cover any variety, or grouping of varieties, of a language (ie dialects, idiolects, registers, genres, jargons, etc). But the term "dialect" is a technical term within linguistics.