I just checked the "Reader's Service Horologion" website which reflects Russian practice, and here is what it says:
And, regardless of the rank of the service, we sing “Lord, I have cried” in the tone of the first sticheron appointed
You've hit it right there: Lord, I have cried is always chanted in the mode of the 1st following hymn.
So the question is: which is the first following hymn? The answer is complex and simple at the same time. Here are some absolutes that apply:
1. On feasts of the Lord and Theotokos (so-called "Despotic Feasts" in Greek - "Despotic" meaning "Master"), only hymns for the feast are chanted.
2. On Sundays (if not #1), the first hymns are in the Mode of the week.
3. During Great Lent and Holy Week (if not #1 or #2), the hymns of the Triodion are chanted first, followed by the hymns for the Saint.
4. On major Saint days (if not #3), only hymns for the saint are chanted.
5. On "regular" days (of not #4), the first hymns are for the day (of the week) in the Mode of the week, followed by hymns for the Saint (if any).
The last question would then be, what defines a "major" Saint day? I don't remember the formula the Typikon uses, but it's basically something like "if the Saint has the full set of hymns" including Liti and OT readings. Someone should look it up and post the exact reference.