This mower, the second one you pictured has the Briggs & Stratton and should be a j19lm or rj19lm. Either one will work just fine. There is also an RJ 19LMC which just stands for copper which is fine too.
The first one you pictured is the Suzuki and it will have possibly even a finer thread but it will be a longer reach plug.
Regardless, you can test for spark and set the gap and it's highly unlikely to be a plug anyways. It's pretty much never a plug on an engine won't start or run.
It does happen but in my experience of thousands of mowers, it's about 1 and every 450 at best so you can always assume that it's not the plug and going to the plug first thing or pulling out or even checking it is almost always a waste of time.
It's almost always fuel so the first thing you should do to troubleshoot is to pop the air filter out and give it a squirt of spray carb cleaner in the hole for an external fuel source and then pull it a few times.
Technically, you should check the oil and make sure it has something that looks like oil in it and it's in the safe area on the stick before you do this. Lol