Đevanāgarī or Nāgarī
Nāgarī (नागरी) and Đevanāgarī (देवनागरी) both the words are used to address the same script. This script - 'ĐevaNāgarī' is used for writing Hinđī, Sanskṛiŧ, Marāthī, Nepālī and many other scripts of South Asia (particularly India). Which word (Nāgarī or ĐevaNāgarī) should be used can not be a subject of debate!
Few scholars use the word 'Nāgarī' to represent an older form of the script. I don't agree. In most of the dictionaries both the words are defined to be synonyms.
The information given here is not necessary for someone who wish to learn Nāgarī. If you like to know more about Nāgarī/Hinđī read on...
'Kāshī Nāgarī Prachāriṇī Sabhā' was a great (but small) organisation. It was formed for the promotion of the ĐevaNāgarī script and Hinđī. At the end of 19th century the 'Sabhā' resolved to publish a big Hinđī dictionary in Nāgarī. Their research team collected a lot of Hinđī words (over 20 years!) from various sources. The result was 'Shabđa Sāgar' (शब्द सागर / sea of words).
The research team was led by the editor Shyām-sunđar-đās. There were five assistant editors - Bāl-kriṣhṇa Bhatta, Rām-chanđra Shuklā, Amīr-Sin̐h, Jag-mohan Varmā and Bhagvān-đīn. Apart from this team there were others who collected words from different source. Particularly Munshī Rām-laganlāl collected lots of words by talking to people involved in different trades.
By any measure, 'Shabđa Sāgar', is a great dictionary of Hinđī. Its second edition was published in 1924. The second edition contains 93115 words in 3999 pages spanning eight volumes. It also contains lot of information about Nāgarī and Hinđī.
As 'Kāshī Nāgarī Prachāriṇī Sabhā' used the word 'Nāgarī' instead of 'ĐevaNāgarī' in its title, we can say that the word 'Nāgarī' was in use in the late 19th century and in the beginning of 20th century.
According to 'Shabđa Sāgar' also, both the words are synonyms. In 'Shabđa Sāgar' under the word 'Nāgarī' they mentioned to refer to 'ĐevaNāgarī'. The script was described under the word 'ĐevaNāgarī' in great details and every time they used the word 'Nāgarī'. This dictionary also published Shri Ozha's chart showing evolution of Nāgarī from Brāhmī (the ancient script of India).
The word Nāgarī (नागरी) seems to have been derived from the word 'Nāgar' (नागर). According to 'Shabđa Sāgar', 'Nāgar' means 'related to city' (and probably 'in the city').
IMHO, as information and knowledge is the core of a civilization and a script is used to preserve it. So, the word Nāgarī might have been used for a meaning like 'a thing related to civilization'.
In some dictionaries 'Nāgarī' is described simply as 'script of city'. City is Nagar (नगर) and not Nāgar (नागर). In Sanskṛiŧ & Hinđī, Dev means 'God' or something divine. The Oxford Concise Dictionary describes ĐevaNāgarī as 'script of divine city'. I don't agree. IMHO, the word 'Nāgarī' is older and 'Đev' was added afterwards (just) to show respect!
Some relate this word with 'Nāgar Brāhmaṇ/s' (a sub-caste of priests).
To read about other alternate spellings like devnagari click here.
Please let me know if I made any mistake in the above text. This text is written by me without any help from any web-page, book or article except the second edition of the 'Shabđa Sāgar'. - Akhilesh Gupŧā