These are called ನಿರ್ವಚನ (nirvacana) or ವ್ಯುತ್ಪತ್ತಿ (vyutpatti) — the tradition going back to Yāska's Nirukta, where a word's meaning is unpacked from its own dhātu. The formula "X iti Y" is itself the classical style: ಅನ್ವರ್ಥಸಂಜ್ಞಾ, a name that carries its meaning on its face. Here's a collection:

ತತ್ತ್ವ / ವೇದಾಂತ

• ಬೃಹತ್ವಾತ್ ಬೃಂಹಣತ್ವಾತ್ ಚ ಬ್ರಹ್ಮ — it is vast, and it makes vast (√ಬೃಹ್)
• ಅತತಿ ಸತತಂ ಇತಿ ಆತ್ಮಾ — that which moves/pervades constantly (√ಅತ್; Śaṅkara also gives ಆಪ್ನೋತಿ, ಅತ್ತಿ, ಅಯತೇ)
• ಪುರಿ ಶೇತೇ ಇತಿ ಪುರುಷಃ — the one who dwells in the city (of the body)
• ಸಂಸರತಿ ಇತಿ ಸಂಸಾರಃ — that which keeps flowing on (√ಸೃ)
• ಶೀರ್ಯತೇ ಇತಿ ಶರೀರಮ್ — that which decays (√ಶೄ)
• ಈಶತೇ ಇತಿ ಈಶ್ವರಃ — the one who rules
• ದ್ಯೋತತೇ ಇತಿ ದೇವಃ — that which shines (√ದಿವ್)
• ಹೃದಿ ಅಯಂ ಇತಿ ಹೃದಯಮ್ — "this one is in the heart" (Chāndogya 8.3.3's famous folk etymology)

ಮನಸ್ / ಜ್ಞಾನ

• ಮನುತೇ ಇತಿ ಮನಃ — that which thinks
• ಮನನಾತ್ ಮನುಷ್ಯಃ — man, because he reflects
• ಬುಧ್ಯತೇ ಅನಯಾ ಇತಿ ಬುದ್ಧಿಃ — that by which one awakens/knows
• ವೇತ್ತಿ ಅನೇನ ಇತಿ ವೇದಃ — that by which one knows

ರಕ್ಷಣ-ಜೋಡಿ ನಿರ್ವಚನಗಳು (the elegant double etymologies)

• ಮನನಾತ್ ತ್ರಾಣನಾತ್ ಚ ಮಂತ್ರಃ — it protects the one who reflects on it
• ಶಾಸನಾತ್ ತ್ರಾಣನಾತ್ ಚ ಶಾಸ್ತ್ರಮ್ — it instructs and it protects
• ತನೋತಿ ತ್ರಾಯತೇ ಚ ಇತಿ ತಂತ್ರಮ್ — it expands and it saves
• ಗಾಯಂತಂ ತ್ರಾಯತೇ ಇತಿ ಗಾಯತ್ರೀ — it protects the one who sings it

ಪ್ರಕೃತಿ (these are the charming picture-words)

• ಖೇ ಗಚ್ಛತಿ ಇತಿ ಖಗಃ — the sky-goer (bird)
• ಉರಸಾ ಗಚ್ಛತಿ ಇತಿ ಉರಗಃ — the one who walks on its chest (snake)
• ಪಾದೈಃ ಪಿಬತಿ ಇತಿ ಪಾದಪಃ — the one who drinks through its feet (tree)
• ಪಂಕೇ ಜಾಯತೇ ಇತಿ ಪಂಕಜಮ್ — born in the mud (lotus)
• ಸರತಿ ಇತಿ ಸರಿತ್ — that which flows (river)
• ದ್ವಿಃ ಜಾಯತೇ ಇತಿ ದ್ವಿಜಃ — twice-born (bird from egg, brahmin from upanayana, tooth!)

ನಾಮ-ನಿರ್ವಚನ

• ರಮಂತೇ ಯೋಗಿನಃ ಅಸ್ಮಿನ್ ಇತಿ ರಾಮಃ — that in which yogis delight
• ಕರ್ಷತಿ ಸರ್ವಂ ಇತಿ ಕೃಷ್ಣಃ — the one who draws everything to himself
• ವಸತಿ ಸರ್ವತ್ರ ಇತಿ ವಾಸುದೇವಃ — the one who dwells everywhere

And one for your Carnatic side: ರಂಜಯತಿ ಇತಿ ರಾಗಃ — that which colours (the mind). Matanga's Bṛhaddeśī gives it fully: ರಂಜಯತಿ ಇತಿ ರಾಗಃ — the raga is that which delights.

A fun technical note: grammarians distinguish words like ಪಂಕಜ as ಯೋಗರೂಢ — the etymology says "mud-born" (which would include frogs!), but convention (ರೂಢಿ) restricts it to lotus alone. So vyutpatti gives the why of a word, rūḍhi gives the what.