Why would any non-Russian Greek Catholic use tri-bar cross?
Why wouldn't?
I thought it's a Russian tradition. Just wondering why non-Russians would adopt Russian traditions.
Only Russian? You sure?
I do not recall from where I read what I am about to write and I'm interested in feedback, but I thought the 3 bared cross was the traditional Eastern Orthodox Christian cross. I read that after the Greek revolution of 1833, the principals of the protectorate powers (the United Kingdom, France, and Russia) were dominated by Protestants and to a lesser extent, Roman Catholics, and they encouraged the use of the cross style used by the Western Churches, which carried over to Constantinople, Antioch, and Alexandria. Even today, manufacturers in Greece frequently use the 3 bared cross on ecclesiastical supplies used in church sanctuaries, i.e. such as the censer, the paten and chalice, candelabra, etc. (Those powers with Protestant influence also pushed the Church of Greece to declare "autocephaly," likewise, and they encouraged the Western influence in Greek iconography of that era.)
1. The Greek Revolution dates from 1821, not 1833.

2. The cross known as the Greek cross looks like a plus sign, with all arms of equal length. It also features on the old post-independence Greek flag (a square flag of a white cross on a blue field), and on the present-day national flag which bears the stripes as well as the cross in the upper left corner. The Roman cross, of shorter horizontal arms on a longer vertical, greatly predates the emergence of protestantism, and, IIRC, was in use in the earliest post-toleration period.
3. Greek manufacturers of church paraphernalia cater for non-Greek customers outside of Greece, including Slavic parishes and monasteries in the diaspora. The three-bar cross is also seen often enough in the northern Greek provinces, where Slavic influences have colored many cultural practices, including spoken dialects, baptismal names, and church praxis.
Even where I live, there is a Greek church whose interior was fully painted with icons about 25 years ago. The iconographers were three brothers from Thessaloniki. In the mural icon of the Resurrection, an angel is seen flying above Christ, holding a cross. A three-bar Slavic cross.