If you wanted someone to help you explain politely, perahps you could have asked politely - by using the traditional term 'please'? 
I'm a bit shocked that you can't see what the problem is with being late in those two instances, but I will explain.
Museum -
I spend a lot of time on my own. I could go and walk around a museum by myself if that's what I wanted, but if I choose to meet up with a friend, then I'm doing so for both the company and the activity. I'd want the companionship of chatting with someone as we went around together - you don't get this by yourself, even if there is a lot to see. I wouldn't mind if it was a rare occurrance, but if you were continually late for me then I'd be cross. Sometimes, I'd much rather make the meeting time 40 mins later than be on my own for that time. I'd rather use that 40 mins doing something of MY choosing - not agreeing to do one thing then finding that someone else is effectively unilateraly, and without prior notice, changing the plan to suit them.
It would also leave me in a predicament - I wander round for 40 mins, you turn up and later want to look at the parts I'd already seen. Do I do it a second time and be bored and resentful that I'd rather be spending the time doing something else, or do I tell you no, sorry, do that on your own after I've left - and risk looking petty.
Or I could go and have a coffee. But maybe I've budgeted for that day, and having a coffee on my own means I can't then have a coffee with you later.
Work -
You said it's a rigid start time. You signed a contract agreeing to the company hours. It is legally not your right to effectively change the terms and make your start time different. Doesn't matter whether it's 5 mins. A rigid time is often there for a reason - because the business needs to run to certain times. It pissed a lot of parents off at my DD's old nursery when some staff were consisently late, because some of us wanted to physically hand our babies over to someone, yet staff weren't there to do that to. Which also meant that staff:children ratios were not as they were supposed to be.
Sure, you make up the time by staying late. But it's selfish to unilaterally decide to think that because you do overtime, that makes up for your consistent lateness. It doesn't - they are two separate issues.
Working overtime does not (necessarily) mean that you are at work at 8:30am as is expected of you, so it does not compensate. If everyone decided to come in late, it might have a really negative impact on the smooth running of your workplace. If you really can't make it in on time, why at least not try and negotiate an 8:35pm start with your employer?