TLDR: You cannot determine where you are taxed solely on the basis of your place of residence and your place of work. If you are taxable in France, Article 10 of the French General Tax Code determines where tax is assessed in France based on your residence. If you are an employee posted abroad under the conditions of Article 81 A, a French exemption may apply under certain conditions.
The place of taxation cannot be inferred automatically
Living in France and working abroad is not, by itself, sufficient to determine in which country your income tax is due.
If you are taxable in France and have a single residence there, tax is assessed at the place of that residence. If you have several residences in France, it is assessed at the place where you are deemed to have your principal establishment. This rule specifies where tax is assessed in France; it does not, by itself, make it possible to conclude that your salary is taxable in France rather than abroad.
When Article 81 A may apply
Article 81 A concerns persons domiciled in France within the meaning of Article 4 B of the French General Tax Code, who carry out salaried employment and are sent by their employer to a country other than France and the country where that employer is established.
Within this framework, salaries corresponding to the activity carried out in the country of assignment may qualify for an income tax exemption. You therefore cannot extend this regime to anyone who lives in France and works abroad.
The employer must be established in France, in another Member State of the European Union, or in another State party to the Agreement on the European Economic Area that has concluded with France an agreement on administrative assistance to combat tax fraud and evasion.
The conditions for the exemption
The full exemption may apply if you can demonstrate that the remuneration concerned was effectively subject, in the country where you carry out your activity, to income tax at least equal to two thirds of the tax that would have been due in France on the same taxable basis.
To make this comparison, you must apply the French rules for determining the taxable basis and family quotient, disregarding any other income received by you or by members of your family. The amount of tax paid abroad constitutes the other term of the comparison.
For certain activities specifically listed in Article 81 A, another condition may be met: the activity must have been carried out for more than 183 days during a period of twelve consecutive months. This condition concerns, in particular, certain construction sites, the installation or operation of industrial complexes, the research or extraction of natural resources, and navigation. It is not a minimum period applicable to all work carried out abroad.
What you can conclude
You must first determine whether you are taxable in France and whether your situation falls within the specific regime of Article 81 A. The mere existence of employment abroad is not sufficient to establish an exemption in France or to automatically determine the country in which tax is due.
If you claim the exemption based on taxation abroad, you must be able to demonstrate that the remuneration was effectively subject to tax and that the two-thirds threshold was met. In principle, proof may be provided by means of a claim within the ordinary statutory time limits; the available information does not specify the concrete documents to be attached or the deadlines in figures.