I think I've found the answer in Chapter 7: The Disputed India–China
Boundary 1948–1960 of the book War and Peace in Modern India by Srinath Raghavan, although the following information can be found in other sources as well.
The northeastern boundary of the princely state of Jammu and Kashmir is called the Ardagh–Johnson Line, which was first drawn in 1865 by William Johnson and proposed in 1897 by John Ardagh. The line was rejected at the time by the (British) Indian government as being "too ambitious", and the Macartney–MacDonald Line, a line that gave more territory to China, ended up being proposed instead to the Qing in 1899. The Qing didn't respond, however, and after the 1911 Revolution, leading to the collapse of the Qing dynasty, the Indian government reverted back to the Ardagh–Johnson Line, which would remain the boundary up to Indian independence in 1947. Nevertheless, the border was considered "undefined" because of a lack of agreement with China, explaining the 1946 National Geographic map.
After independence, Prime Minister Jawaharlal Nehru stated in 1950 that the border from Ladakh to Nepal was defined "chiefly by long usage and custom", as opposed to his emphasis on the McMahon Line to the east that delineated Arunachal Pradesh — evidently, "India was surer of its rights in the eastern sector [of its border with China] than in the
west." India didn't want to push border issues with China too far since it "was far from consolidating its influence in the
border regions and ill-prepared to counter any efforts by China to take
possession of these parts."
In 1953, India decided to make official maps to show a defined border with China. To quote the book,
The crucial decision, in retrospect, lay in the
Ladakh sector. Here Delhi decided neither on the ambitious Ardagh
Line nor on the MacDonald Line, but a "compromise line which had
some plausibility." This line placed Aksai Chin within Indian territory. The foreign secretary at the time, R.K. Nehru, later recalled that
"in 1953, our experts had advised us that our claim to Aksai Chin was
not too strong." The prime minister was "agreeable" to adjustments in
"Aksai Chin and one or two other places" being made "as part of a satisfactory overall settlement."
Nehru either didn't bring up or didn't negotiate this boundary issue in later meetings with the Chinese government, including in the 1954 Sino-Indian Agreement and during talks with Premier Zhou Enlai. There were, of course, border disputes, ranging from China's construction of a highway passing through Aksai Chin to the 1962 Sino-Indian War.
Unlike earlier maps, like this one from 1947, Indian maps published during the 1950s, such as this one from 1954 and the one below from 1959, align with India's current territorial claims:
So essentially, the current border dates back to the 1950s and is a modification of a line drawn by the British in the late 1800s that was never really that well defined or enforced.