The International Federation of Red Cross and Red Crescent Societies does indeed use a similar flag from time to time, as I have seen it around here, mainly on trucks carrying humanitarian help. However, there is never the red border around it. More often a full logo of the organization is used on a white bed sheet - the red crescent and cross in the middle of two black concentric circles, between which is written the name of the organization.
On the headquarters of the organization here in Zagreb the two more usual flags (Red Cross flag and Red Crescent flag) are hoisted permanently.
The ICRC website explains:
The International Movement of the Red Cross is composed by the International Committee of the Red Cross, the International Federation of Red Cross and Red Crescent Societies and the National Red Cross and Red Crescent Societies.
The ICRC is exclusively Swiss (thereby giving the whole Red Cross Movement its neutrality), while the IFRC is made up of societies from 188 nations and territories. The ICRC is concerned with upholding the Geneva Convention in times of war, and to render aid in war zones, while the IFRC has a broader scope, including natural disasters relief, aiding refugees, and blood collection.
From the document "History of the Emblem" (formerly posted on the ICRC website), dated 1 November 1994:
1864 - The first-ever Geneva Convention was adopted: the red cross on a white ground was officially recognized as the distinctive sign of the medical services of armed forces.
1876 - During the Russo-Turkish war, fought in the Balkans, the Ottoman Empire decided to use a red crescent on a white ground in place of the red cross. Egypt also opted for the red crescent and Persia subsequently chose a red lion and sun on a white ground. These States made reservations to the Conventions, and their exceptional signs were then written into the 1929 Conventions.
1949 - Article 38 of the First Geneva Convention of 1949 confirmed the emblems of the red cross, the red crescent and the red lion and sun on a white ground as the protective signs for army medical services. It thus excluded the use of any exceptional sign other than the red crescent and the red lion and sun.
1980 - The Islamic Republic of Iran decided to give up the red lion and sun and use the red crescent in its place.
1982 - The International Federation of Red Cross and Red Crescent Societies adopted as its emblem the red cross and red crescent on a white ground.
So the Red Sun & Lion were evidently official; but for further clarification regarding Magen David Adom, I quote this from the document "The Red Cross and Red Crescent Emblems," dated 1 September 1989:
The emblem of the red shield of David is covered in a reservation whose validity has been challenged by a number of authors. Without embarking on a lengthy analysis of a controversial technical legal point, we hold the view that opponents of the State of Israel are bound to respect Israeli medical personnel and equipment on the field of battle.
In any event, the protective emblem is not constitutive of protection under the Convention; it is merely the visible sign thereof. Members of the medical service shall command respect by virtue of their relief mission, and not because they are indicated by any given distinctive sign.
On the other hand, the International Committee of the Red Cross has been unable formally to recognize the Israeli Red Shield of David Society (Magen David Adom), with which it has maintained excellent working relations for over forty years, owing to the fact that the Society does not fulfil one of the conditions for recognition of new National Societies laid down by the Seventeenth International Conference of the Red Cross in Stockholm in 1948 and confirmed by the Twenty-fifth Conference in Geneva in 1986, to the effect that the applicant Society, to be entitled to recognition, must "use the name and emblem of the Red Cross or Red Crescent in conformity with the Geneva Conventions". For the same reason, the League of Red Cross and Red Crescent Societies could not admit the Israeli relief society.
Incidentally, there was a reference to a petition from India in 1977, requesting the acceptance of still another symbol (which was not identified). This request was also denied.
Francois Bugnion, in The Emblem of the Red Cross: A brief history, Geneva: International Committee of the Red Cross, 1977, p. 65, presented a number of other suggestions and proposals:
Afghanistan: Red Archway (emblem from the flag), application for recognition filed in 1935, but rejected.
Congo: Red Lamb, one of several rival societies that emerged after the independence of Congo (later Zaire, now Democratic Republic of the Congo), active in 1963 and 1964.
India: Red wheel, discussed in the Indian Red Cross society after in the time after independence, but abandoned.
Japan: red sun and red strip, emblem of Hakuaisha society, founded 1877 and equivalent of a national red cross society, society changed name to the Red Cross and adopted the red cross emblem in 1866.
Lebanon: Red cedar, suggestion only given attention in "preliminary discussions" in the post-war years.
Sudan: red rhinoceros, suggested as a common emblem for the new Sudanese society which united the previous branches of the Red Crescent Society of Egypt and the British Red Cross. In the end, the Sudanese society chose the red crescent.
Syria: Red palm, suggested after World War II, as a common "koranic and biblical emblem." The suggestion was rejected.
Thailand: Red cross and flame. Emblem of the Sabha Unalome Deng relief society founded in 1893, a combination of Buddhist emblems with the traditional red cross. Recognition for emblem sought in 1899 and 1906, but rejected. Thai society adopted red cross in 1906.
Other "mixed" symbols suggested include a Red Palm for Syria and a Red Rhinoceros for Sudan, along with a Red Archway for Afghanistan and a Red Lamb for Congo. In Asia, there's a Red Wheel for India (along with a Red Swastika, also for Sri Lanka), a Red Sun and Strip for Japan, and a Red Cross and Flame for Thailand.