Charging

Where is this five volts coming from? The engine cannot start or run with the battery voltage being that low. You cannot have thirteen volts on the battery either if the engine is not running. You might be measuring a "surface charge" if the battery was just being charged. Otherwise a fully-charged battery will read 12.6 volts.

What symptom are you trying to solve?

I cannot quite see for sure, but it looks like the negative probe is on the battery's negative cable. If that is right, there is something wrong with your meter. I have about fifteen digital meters from my TV repair days, and one of them reads way wrong on all of the "DC Volts" scales. Find another battery to measure. If that one reads correctly, there is a film of black corrosion on one of the terminals in your photo that is interfering with the reading. It helps to poke really hard on the probes to get them to cut into good clean lead.

That battery should measure 12.6 volts if it's good and fully-charged. If it is good but discharged, it will read closer to 12.2 volts. If you find it at around 11 volts or less, it has a shorted cell and must be replaced.

To tell if the charging system is working, measure the battery voltage again with the engine running. It must be between 13.75 and 14.75 volts. I suspect that is what you have since you are not having a cranking problem.