In reality, all the three names are more or less interchangeable. Twin-towns, partner-cities, and sister-cities are the ones connected with friendly relations – and recurring joint projects organized on their basis. In other words, such can be called any two towns that consider each other a little bit more of a family than the others, and that cooperate in a wide range of spheres as equal partners. In continental English, it is a little more common to say "twin-towns", whereas the US variant favors the term "sister-cities".

Nowadays, the establishment of such friendly relations between cities is usually heralded in a written agreement. However, it was only in 2019 that the Russian ministry of Foreign Affairs provided recommendations for a more or less standardized form of such a document. And let us be reminded that the international twinning movement was born out of a civic initiative rather than was initiated by politicians and lawyers. This means that the legal part of the question has not been addressed for a rather long time.

The first ever agreement on the establishment of the "bond of friendship" between Stalingrad and Coventry was but a brief newspaper article, and yet after this, both cities wholeheartedly considered themselves properly twinned. The announcement was made publicly, after all, with both sides being worthy of every piece of trust!

Nowadays, such a document can still be called simply the "Agreement on Friendship and Cooperation". Which means that its parties may call themselves whatever they like.

Of course, there are certain unwritten rules, in Russia at least. For example, Moscow believes that the cities within the same country can only become partners, whereas the ones in different states – only twins. Stalingrad-Volgograd used to attribute the title of partners to the cities from Russia and the ex-Soviet republics, and the title of twins – to the ones located farther abroad.

Of course, there are exceptions to every rule. For example, the Turkish Gaziemir initially made an agreement not with Volgograd as a whole, but with one of its districts, and so remained in the list of "partner-cities". The Armenian Yerevan, on the other hand, became Volgograd's twin-town simply because suggested that the respective document be called "a Twinning Agreement" – which was and is perfectly legit!