Weeds in lawn

Hi everyone,
Our lawn is due for its first cut soon. We seem to have a collection of a couple of weeds and I have no idea if they're the annual type which will get lost with a cut, or if I need to try to dig them out. Here are the photos of each:
The first one seems the most vigorous and is in quite a large patch, as well as dotted about, so I wouldn't be surprised if this requires digging out. The second one is dotted around, and the flowers are quite nice, but I'm not sure if it's one to dig out or not.
Thanks for any help,
Lucid
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Thanks Verdun. We're having to do some further levelling and seeding so if they need to be pulled out I guess it won't be too bad. Are they perennials then do you know? I try to avoid weed killers for the wildlife, or are there any that are ok?
Thanks,
Lucid
You could just leave them.... As you say, they are pretty. There really aren't any wildlife friendly herbicides. I think it is self-heal (the first one) and speedwell. I LOVE speedwell and would happily have more in my lawn. Flowering lawns are quite nice for wildlife and can be pretty. http://www.theguardian.com/lifeandstyle/2015/jul/18/flowering-lawns-gardens-alys-fowler Self-heal I'm not crazy about but it doesn't do any harm. You can dig it out and reseed grass.
Try an artificial lawn
only joking Verdun 
Just pointing out that there are different ways of looking at things. Here's the label of a common "weed and feed" https://www.lovethegarden.com/file/evergreen-complete-4-1-labelpdf Note on the left where it says "Harmful to aquatic organisms." And that's not just the herbicide. All that nitrogen in the "feed" goes into waterways.
http://learn.eartheasy.com/2010/06/6-reasons-to-avoid-using-%E2%80%9Cweed-%E2%80%98n-feed%E2%80%9D-on-your-lawn/
I like my lawn flowers, but if Lucid doesn't want any, I think digging them out would be preferable, especially with the new pond!
Agree with Watery, no need for weed killers, and certainly not as a first choice solution.
A lawn is a living thing. It doesn't need to be perfect unless you're planning to charge people to play crown green bowls on it. Dig the weeds out according to time and inclination, and the grass will spread and grow back in to it.
You can always reseed patches if the initial weed removal has left big enough gaps.
'mine is low maintenance and cut twice a week' sounds like a contradiction in terms. I'm pretty sure most people would not see cutting their lawn twice a week as low maintenance.
Personally, I would also take in to account my electricity bill and carbon footprint as well as the environmental impact of using unnecessary weedkiller.
And all credit to OP Lucid for trying to avoid weedkillers.