I noticed Monier Williams has entries for both om and aum. For the latter, he defines it as "the sacred syllable of the Shūdras". What particular association does aum have with Shūdras or is it a misinformed definition?
MW is quoting from PWG, which is quoting from ŚKD, which is quoting from (Bṛhat)Tantrasāra, which is quoting from Kālikā-purāṇa.
In KP, ch. 56(58) we can read:
mantrāṇāṃ praṇavaḥ setus tat setuḥ praṇavaḥ smṛtaḥ |
kṣaraty anoṅkṛtaḥ pūrvaṃ parastāc ca viśīryate || 72/70
Praṇava (
Om) is a dam for mantras, therefore praṇava is known as dam.
Without making
Om before, it flows away, and [without it] at the end - it perishes.
niḥsetu ca yathā toyaṃ kṣaṇān nimnaṃ prasarpati |
mantras tathaiva niḥsetuḥ kṣaṇāt kṣarati yajjvanām || 77/75
And like in a flash the water without a dam flows downwards,
similarly in a flash a mantra without a dam flows away from sacrificers.
caturdaśasvaro yo'sau śeṣa aukārasaṃjñakaḥ |
sa cānusvāracandrābhyāṃ śūdrāṇāṃ setur ucyate || 76/74
Which is the 14th vowel, the last [vowel], termed 'sound
Au',
this [vowel] with candrabindu (with semicircle and dot) is said to be dam for śūdras.
Few more quotations from somewhere:
svāhāpraṇavasaṃyuktaṃ mantraṃ śūdre dadaddvijaḥ |
śūdro nirayagāmī syādvipraḥ śūdratvamāpnuyāt || (devīyāmale)
[When] a brahmin is giving a mantra together with
svāhā [and/or] praṇava (
Om) to a śūdra,
then the śūdra will go to hell and the brahmin will turn himself into śūdra.
śūdrasya praṇavasthale caturdaśasvaro vindusaṃyuktaḥ |
For a śūdra the 14th vowel (
Au) with [candra]bindu takes place of praṇava (
Om).
mantrāṇāṃ praṇavaḥ seturdvijānāṃ parikīrttitaḥ |
caturdaśasvaro'nyeṣāṃ candrānusvārasaṃyutaḥ |
Praṇava is known as a dam for mantras for twice-born,
[while] for others it is the 14th vowel (
Au) together with candrabindu.
So, there are some texts which in the case of śūdras recommend to use औँ Aum̐ instead of ओँ Om̐. These are two different bījas, that´s why Monier has both in his dictionary.
(NB! It is different from the illiterate NewAge tradition to spell OM as AUM. By sandhi rules अ A + उ U + म् M give always ओम् OM, not AUM.)