I entered "Happy Birthday" in two different online translators. Both give ਜਨਮਦਿਨ ਮੁਬਾਰਕ, but one transliterates as janamdin mubarak and the other as janamadina mubāraka. Does that reflect two different varieties?
The pronunciation and transliteration is janamdin mubārak. This takes into account the phenomenon of schwa deletion. See the following: Schwa deletion in Indo-Aryan languages - Wikipedia. Some transliteration systems may not factor in schwa deletion and thus give the transliteration as janamadina mubāraka, even though the pronunciation is janamdin mubārak.
I was going to disagree with you at first, Dib, but after looking into it a tiny bit I think you're probably quite right.
However, in my experience, most transliteration schemes (so-called) would, I think, make some provision for reflecting the spelling conventions of the original language. What I mean is, I don't think there are many "transliteration" schemes that would use such a misleadingly pedantic transliteration as janamadina mubāraka, I think they would generally just go with janamdin mubārak, because it's just so much more useful. Whereas, I would say that most transliteration schemes would render ਬਹੁਤ as bahut and that rendering it as bot or bót would be transcription as it completely obscures the original spelling.
So in short I think it's probably fine to call janamdin mubārak a transliteration, although I do agree it's probably a slightly loose application of the term.
A strict transliteration is where a grapheme in one system is consistently represented by a grapheme (which may be a digraph or trigraph) in another. I know little about the scripts used to write the languages of India (other than roman!) but I do know that Gurmukhi does not have a vowel cancellation sign like the virama in Devanagari. When transliterating Devanagari the obvious thing to do is simply to omit the implied vowel which has been cancelled. The question is whether when transliterating Gurmukhi the absence of a vowel cancellation sign should be ignored.
There is not necessarily a hard and fast distinction between transliteration and transcription. Many transliterations go in for a bit of compromise, often to compensate for a lack of symbols in the target script. A transliteration should be such that anyone familiar with how to write the language being transliterated and the rules being employed knows exactly what the original is.