There are four
named conditional patterns.
Zero -- IF present, present.
First -- IF present, WILL verb.
Second -- IF past, WOULD verb.
Third -- IF HAD past participle, WOULD HAVE past participle.
There are many variants of these patterns, but they have no names!
Because could can mean would be able to, and could have can mean would have been able to, could may substitute for would in the second and third conditionals.
The backshift of the Zero Conditional has no name. (IF past, past.)
If they expected to minister to others, they had to look after their own fellowship with God.
The pattern with would (or could) in both clauses has no name. The use of would or could in the if clause is unusual, and they are only used when emphasizing that the fulfillment of the condition is in the interest of the listener (or possibly the speaker). There's an element of politeness brought in by the use of would or could in these cases.
If you would/could bring cake to the party, I would/could bring cookies.
CJ