As a CPA I would offer the following advice:
Your obligation to the state of New York, if any, is a facts and circumstances question. If you reside in DC and do a majority of your work there AND you only take occasional trips to New York for meetings or training, etc. - You have no obligation to the state of New York.
The New York "Convenience of the Employer" rule and "State Reciprocal Tax Agreements" are all drawn with the situation where someone lives in, say New Jersey, but routinely works in New York. The federal courts have weighed in on this issue a number of times. The bottom line has been wherever you butt is sitting when you do the work, that is where you are going to be taxed!
Even if you occasionally go to New York and stay a couple of days rewriting code, the courts have considered this "de minimus" and thereby making you not subject to New York tax.
WARNING: Occasionally employers misinterpret these events and withhold income tax in their state and not the state in which you reside. It is your duty, and right, you object to that happening and you should do so to protect your interests.
Based upon the facts as you have described them, your employer must withhold DC income tax. If your employer mistakenly withholds New York state income tax, DC can and will levy your employer for the DC state income tax that should have been withheld AND for the state (in the case district) unemployment insurance (tax) they should have paid on your behalf.
There are three reasons this is so important to you:
You should not be burdened with filing a New York state income tax return to get back New York state income tax that was withheld from your pay check(s). Plus, having to file a non-resident New York state income tax return puts you "in the system." Trust me, you do not want to get into a state's income tax system unless you are required to do so. The courts are replete with cases where a resident of one state got tangled up with another states tax department and ended up having to go to court to undo liens and levies, etc. If you get it right to state with you will avoid a myriad of possible problems.
You and the District of Columbia have a vested interest in DC tax being properly withheld. You would be subject to an underpayment penalty is you had to first file a non-resident New York return and get back you New York withholding so that you could then pay you DC income tax. (In essence, unless you have plenty of money in the bank, you would have to quickly, in February, file a non-resident New York state income tax return to receive a refund by April 15th so you could pay your DC income tax on the due date. Furthermore, if you owe more than $100 in DC income tax on April 15th, you will be penalized for not having made quarterly estimate payments. Then penalty is 10.00% per annum, compounded daily.
Most importantly, Should you be terminated, you are in "No Man's Land" regarding your ability to receive unemployment benefits. Since your employer did not withhold DC income tax, they would not have paid anything into you DC unemployment benefits account. So you will not be able to receive unemployment benefits in DC. Since you are not a resident of the state of New York, you will not be able to receive unemployment benefits in New York.
Simply put: Make sure your employer classifies you as a DC employee and withholds DC income tax. If not, make sure you keep a great record of events. Should you get into the "hassle of your life" it will all make a great magazine article.