My KPT-post has proved trickier to write than I expected (but it's coming!) so in the meantime, a question that I answered in the Facebook Finnish study group Let's Learn Finnish Language. Kirez, who asked the question, has kindly allowed me to publish his question along with my answer here.
Google translate käännä “puheikielallä” as “speech language”...
I’m guessing a better translation is “vernacular”?
It gives me no translation for “yleiskielallä” - I’m guessing more formal or written language?
"Puhekieli" or spoken language usually refers to standard spoken Finnish (also "yleispuhekieli"). In my opinion, vernacular is not a great translation, as vernacular usually refers to more specific dialects or registers - in Finnish, vernacular is "kansankieli" or "paikallismurre" (local dialect). Like in all spoken varieties of any language, there is no one fixed form of standard spoken Finnish: forms vary depending on the area the speaker is from and the context the language is spoken in, and there isn't a clear cut line between "yleispuhekieli" and more specific registers of the language. However, the forms used in the Helsinki region are often seen as the neutral choice (but obviously it depends on who you ask!).
"Yleiskieli" is indeed a more formal version of Finnish and can be translated as standard Finnish. In its written form, it's also referred to as "kirjakieli". Spoken yleiskieli is used mainly in formal situations (often the speaker is reading out loud or has written and memorized what they're going to say), like the news, scripted shows on radio or tv, plays etc. In its written form yleiskieli is also called "kirjakieli", literally "book language" usually translated as standard written Finnish.