This will be beginner to beginner advice, so take it for what it's worth.
First I do think you are right that you only need a single line cutter for re-cutting. The angle of the cutter might depend on what kind of points you are trying to cut.
As for tools, I recently bought an Ullman single line cutter and it is outstanding. I also appreciate that you can buy directly from Michael Ullman at this website, and the tool arrives in a simple USPS priority mail box from the manufacturer. His tools are certainly not the cheapest, but i believe they are worth the extra cost.
https://www.ullmanprecision.com/products/
