Stressed syllabic r and syllable onset r are rounded, commonly at least, as in "bird", "bred". Unstressed syllabic r and syllable offset r are unrounded, as in "better", "lard". This is for an r-ful dialect. I don't speak an r-less dialect, but I suppose in those dialects, there also wouldn't be rounding in unstressed or offset position of whatever replaces r.
The transcriptions you see in dictionaries imply that (what I call) syllabic r is some sort of diphthong. I don't hear any more change in quality during the articulation of syllabic r than for any other non-diphthongal vowel, so, phonetically, those transcriptions don't make any sense, to me. There have been phonemic theories of English vowels that try to place them all in a system with simple vowels matched up somehow with diphthongs, and probably, historically, this has played some role in these strange dictionary transcriptions.
What do stressed syllabic and syllable onset r have in common that causes them to work the same way for roundedness? I have an idiosyncratic theory about that, which is a variation of a theory proposed by Saussure in his Course in General Linguistics. Saussure has the idea that language structure has two aspects, the paradigmatic, which concerns what role pronunciations play in comparison to others, and the syntagmatic aspect, which concerns how language elements are pronounced in the stream of speech. (This is a rather free interpretation of what Saussure says, which I may not completely understand.)
Languages have different syllable types, as we know, and within a language, there are difference in syllable structure depending on initial versus final, and so on. This concerns paradigmatic structure. But, correspondingly, we can find in the stream of speech differences in individual sounds that depend on where the sounds come in a syllable. This concerns syntagmatic structure.
Within a syllable, we find that aperture in the mouth is small at the beginning of a syllable, increases to a maximum at syllable peak, then decreases through the syllable offset to the end of the syllable. Looking at the articulation of individual sounds, accordingly, when a sound is in the onset of a syllable, aperture will be increasing during the articulation -- Saussure refers to this as "explosive". At the peak of the syllable, aperture is at a maximum, then in syllable offset, we find sounds during which aperture is decreasing. Saussure calls the sounds in syllable offset "implosive".
What does Saussure say about the sounds of syllables whose aperture doesn't actually increase smoothly through their onsets or decrease smoothly through their offsets, like English "stretch", for instance? He doesn't deal with this difficulty. But I have a proposal about it. Instead of explosive and implosive, suppose that the sounds in the onset of a syllable are stressed and those in syllable offset are unstressed. Arguably, if phonetic features are independent aspects of articulation, as proposed in The Sound Pattern of English, we need to extend the stress feature from vowels to the interpretation of consonant articulation, anyhow.
Getting back to the rounding of r, then, we can now simplify our account in a principled way. It is the stressed r that is rounded, whether in syllable onset or at syllable peak, and the unstressed r is unrounded, whether syllabic or not.