The Hebrews and Greeks did indeed have a word for a person. It is the word we most often see translated as “soul.” 

When the Bible talks about souls it is a reference to persons. For example, when  Peter says 8 souls were saved through water he means 8 persons were saved  through water.  

When Luke writes that 3,000 souls were saved he means 3,000 persons were saved.

The Bible indicates that God is a single soul. He is a single Person.

• Old Testament – Hebrew: nephesh, 1 Samuel 2:35; Psalm 11:5; Proverbs 6:16; Isaiah 1:14; 42:1; Jeremiah 5:9; cf. 5:29; 9:9; 6:8; 12:7; 14:19; 15:1; 32:41; Ezekiel 32:18; Amos 6:8;
• New Testament – Greek: psyche, Matthew 12:18; Hebrews 10:38.

What an unusual and confusing way for a 3-in-1 Person God to refer to Himself! 

Do Trinitarians really expect anyone to believe these are references to a 3-Person Being? 

No, the scriptures always describe one person, one soul, individual as God.