The acid is not optional if you are storing the jars. 1 tbsp bottled lemon juice or 1/4 tsp citric acid per 500 ml jar, doubled for a litre jar, into the empty jar before the sauce. Bottled rather than fresh, because the acidity of a fresh lemon varies.
Reduce it properly. Thin sauce in the jar stays thin. Expect to lose about a third of the volume, and use wide pans rather than one deep one.
Full rolling boil for the whole 40 to 45 minutes. A simmer does not reach the temperature the time assumes. Start the clock when it is boiling hard.
Do not invert the jars. The seal comes from the vacuum as they cool. Check every lid before storing: firm, no flex. Anything that did not seal goes in the fridge or the freezer.
Honest caveat: tested water-bath formulations are plain acidified sauce. The oil and oregano here put it outside the letter of one. To stay strictly inside a tested process, can it plain and add them on opening. Or freeze it, which avoids the question.