The first picture looks like Peppervine to me.
The reason these kind of plants come back is because they have huge rhizomes that serve as energy reserves. Thats why they can come back after the leafy top is destroyed. The best way to deal with it is small weekly repeated chemical applications instead of a yearly nuke. You basically have to wage attrition warfare. Instead of exausting yourself just focus on quick poison application runs.
Snip a limb to expose the inside, iimidetely apply brush poison to the cut, walk away. The plant will literally suck the poison in. This is best done near ground level if you can get near the center mass. Even if it looks dead, keep snipping and painting every week.