No. At most, wording was merely proposed, but did not pass into law.
There are numerous 1913 publications that mention the proposed law.
For example, Railway Journal, April 1913, volume 19, page 10:
The bill failed to pass but here is how one sentence of the bill was worded: "When two trains approach a crossing, both shall stop, and neither shall go ahead until the other had passed by".
See also National Bulletin, Volume 5, page 160, April 1913:
Bill pending in Kansas legislature providing that "when two trains approach a crossing both shall stop, and neither shall go ahead until the other has passed by," could hardly be called progressive legislation.
And Federal Trade Reporter, Volume 2, page 204, April 1913:
"When two trains approach a crossing both shall stop, and neither shall go ahead until the other has passed by," is said to be the wording included in a bill introduced in the Kansas legislature.
The earliest reference I see is page 1 of the 28 March 1913 Wall Street Journal.
Bill pending in Kansas legislature providing that "when two trains approach a crossing both shall stop, and neither shall go ahead until the other has paused by," could hardly be called progressive legislation.
However, (as already mentioned in the OP) there is a February 1915 statement in Railway and Locomotive Engineering, vol. 28, page 40:
The New York Times is responsible for the statement that in an endeavor to straighten out conflicting interests at level crossings in Kansas railroads, a bill has been introduced in the State legislature in which the statement is said to appear that “when two trains approach a railroad crossing, both shall stop and neither shall go ahead until the other has passed by".
So it seems to be entirely bogus. Really, it seems to have originated as a concluding sentence of a politics commentary article in the "Review and Outlook" section of the Wall Street Journal, which wasn't meant to be taken literally, but was reproduced as if true (as if a bill with that language was actually proposed) by many sources, and then later further exaggerated into being an law.
The actual context can be seen from reading the entire Wall Street Journal column:
REVIEW AND OUTLOOK
AN ASPECT OF POLITICS
Doubtless many reasons could be adduced for the political unrest at present so strikingly In evidence. There seems to be a widespread demand for change rather than progress. It is visible in the perversities of decadent art; in music, cacophanous and unmelodious, designed apparently to surprise and startle at any cost; in literature which would have been suppressed by the police a generation ago, and in social conversation upon topics whose mere names would not have been breathed in any mixed assembly.
Our politicians and their pernicious activity might be explained on the theory that the same heady leaven is working in them. But there is another explanation more flattering to hitman vanity, and more promising of good to come. Much of the attitude of the politicians towards the corporations seems to be actuated by no better motive than revenge. Since their irregular profits were cut off, they are endeavoring to indemnify themselves by retaliatory legislation.
No harder blow to the politician's pocket was felt than when the railroad pass became illegal. This was a law which could be enforced, and it has already worked enormous benefit in the purifying of railroad management, which at one time required disinfecting as much as our politicians do now. Corporations also are declining to pay for protection against strike legislation, and a large source of illicit revenue has thereby been cut off. Perhaps the politician does not realize it, but under this heading would come vexatious regulating laws, the costly and unnecessary "full train crew" law, and other measures which will be familiar to everybody.
It is doing no injustice to Albany to say that had the Stock Exchange been willing to resort to means Wall Street considered allowable a generation ago, no restrictive measures would have passed either the Assembly or the Senate. "Lobbying" was unquestionably expected, and not the less so that its results would have meant the introduction of similar legislation every year. The Stock Exchange in this case, as in the case of the stock transfer tax of 1906, has firmly refused to pay ransom.
The situation will cure itself, a generation of politicians will succeed lacking the unholy experience of graft and corruption possessed by their predecessors. The whole country will be the gainer, and for the present, we must possess our souls in patience.
Bill pending in Kansas legislature providing that "when two trains approach a crossing both shall stop, and neither shall go ahead until the other has paused by," could hardly be called progressive legislation.
Furthermore, it was already well established by this time, for example quoting the 1892 US Court of Appeals 8th Circuit case Kansas City, Ft. S. & M. R. Co. v. Stoner:
When two trains approach a crossing at the same time, for safety's sake, the rule is adopted that the one which first reaches and stops at the post upon its line is entitled to precedence in making the crossing
Update:
I found one earlier mention of this supposed law, but with respect to New York rather than Kansas:
In the September 1911 Law Notes, volume 15, page 118:
INTELLIGENT LEGISLATION-As a companion piece to the bill said to have been introduced in the New York legislature, providing that when two trains approach a crossing together both must stop and neither must start again until the other has passed, [goes on to criticize the wording of a totally unrelated Florida law]
Further update:
From the 1909 Minutes of the 25th Annual Meeting of the Association of Edison Electric Companies, at page 395:
[Everett W. Burdett speaking]: This reminds me of a law which is said to have been passed in Massachusetts, in the one year when the “Know-Nothings” had control of the legislature. In order to decrease the number of accidents of the character to which I have referred, the “Know-Nothing Legislature” passed a law to the effect that “When two passenger trains approach each other at a crossing both shall stop, and neither shall proceed until the other has gone over." (Laughter.)
There were actually laws requiring what was referred to as a "know-nothing stop" (named after the Know-Nothing party of Massachusetts). According to the 1882 Van Nostrand's Engineering Magazine, Volume 27, pages 61-63:
The so-called “know-nothing stop” appears to be in force everywhere at points where one track crosses another at grade. In some states this is obligatory by state law. But the practice is universal, and appears not to depend at all upon state law.
But this “know-nothing stop” law is merely the requirement to stop at the crossing rather than cross without stopping. The reference also says this law had already been repealed in Massachusetts as of 1882.
More specifically, from the 1874 Railway Age, Volume 6 at page 350:
In the year 1855 the Legislature of Massachusetts took the matter in hand and passed a law to the effect that all trains approaching a grade crossing should come to a full stop within 500 feet of the crossing, not to start again until a proper signal had been shown. As a majority of the Legislature of that year was of the "Know-Nothing" party, it was called the "Know-Nothing Legislature" and hence the stops caused by that law became known as "Know-Nothing stops."
Link to full text of law, chapter 452 of Act of 1855
