Transcription schemes and their diacritic differences
There is no real standard for the diacritics in reconstructed Proto-Indo-European. There are lots and lots of different transcription schemes, and lots of individual variations on each scheme, which sometimes makes it hard to know exactly what a specific writer means – especially if you look at older texts such as Pokorny’s Indogermanisches Etymologisches Wörterbuch.
The scheme you describe as being used in your book here is the one used in LIV² and (with the minor variation that the labiovelars are transcribed with ⟨ʷ⟩ instead of ⟨u̯⟩) also the one favoured by the Copenhagen school, so it’s quite well-known and widespread. Going from memory, I think this particular scheme originated at Leiden University and in Brill publications, but don’t quote me on that – it’s definitely spread beyond just that now, at any rate.
Another similar variant uses circumflexes instead of inverted breves for palatalisation, writing ⟨k̂⟩ and ⟨ĝ⟩ insteaed of ⟨k̑⟩ and ⟨g̑⟩. This is perhaps more intuitive, since it mirrors established usage in Esperanto. I don’t think I’ve seen that variant used together with ⟨u̯⟩ to write labiovelars.
Wiktionary uses its own scheme, which is very American-influenced. It uses ⟨y⟩ instead of ⟨j⟩ for the palatal semivowel, but otherwise seems identical to the scheme you say you would prefer to use for yourself.
Theoretical reasoning for diacritic choices
Palatals
To my knowledge, there is no particular theoretical reason behind the choice of which diacritic is used to show palatalisation.
Acute accents are commonly used for palatalisation in various modern languages, so they make an obvious and practical parallel; circumflexes are used in Esperanto as mentioned above.
I don’t know of any actual orthographies that use inverted breves for palatalisation, nor do I know why the inverted breve was chosen in the Leiden–LIV–Copenhagen scheme. Possibly just to reduce the number of diacritics required to transcribe PIE to three: acutes (for stress), macrons (for vowel length) and inverted breves (below for semivowels, above for palatalisation). Though using acutes would have had the same effect, so that’s hardly a compelling argument.
Moreover, I’ve always found it a bit odd that probably the most widely used and familiar diacritic for denoting palatalisation appears to be completely absent: the caron. I don’t recall ever seeing a transcription scheme use ⟨ǩ, ǧ⟩ to denote palatovelars in PIE.
Semivowels
Using the inverted breve for the semivowels is more theoretically based, since they do, in Indo-European morphophonology, alternate with full vowels based almost exclusively on syllable structure. Indicating this is useful and highlights the identity of the skeletal phonemes in ablaut patterns like *k̑ei̯- ~ *k̑i- in a way that *ḱej-/*ḱey ~ *ḱi- does not.
This is also the reason why I personally prefer not to use ⟨u̯⟩ to denote labiovelars: these do not contain the semivowel phoneme and do not alternate with full vowels (see note below).
Which to choose?
This is easy to answer:
Unless you’re writing for a publication that requires a specific transcription scheme, there is absolutely nothing wrong with using whichever scheme you find most intuitive, aesthetic and/or easy to type.
A note on horses
The word you quote in the question is actually incorrectly transcribed. The ‘horse’ word is *h₁ék̑u̯os, not **h₁ék̑u̯os. PIE is reconstructed with three series of velars: plain, palatalised and labialised (labiovelar). The transcription ⟨k̑u̯ / ḱʷ⟩ would denote a palatalised and labialised velar, which is not generally recognised. The sequence palatal velar + u̯/w is not common, but it does pop up here and there, most notably in *h₁ék̑u̯os ‘horse’ and *k̑u̯ō(n) ‘dog’.
In the branches where palatovelars remained distinct (satem languages), the reflex of *k̑u̯ generally corresponds to the regular outcome of both elements, one after the other; e.g., Sanskrit aśvaḥ and śvā́- (palatovelar > ś, semivowel > v). In languages where plain and palatalised velars merge, *k̑u̯ and *kʷ largely merge as well; e.g., Latin equus /ˈekʷus/, Germanic *ehwaz (with Germanic Sound Shift). In some cases, the sound changes are parallel, but the biconsonantal nature of the original cluster remains as gemination, e.g., Greek ἵππος (*kʷ before back vowels > p, but doubled here – although the Greek form is unexpected in other ways, so the gemination may not actually relate to the original cluster at all).
The sequence *k̑u̯ also more or less regularly corresponds to *k̑u before consonants, which plain labiovelar *kʷ usually doesn’t; e.g., the genitive of *k̑u̯ō(n) is *k̑unés (disyllabic), as opposed to the genitive of *gʷenh₂ ‘woman’, which is *gʷneh₂s (monosyllabic).
