That looks like the weed to me. There are some ornamental oxalis and a native woodland one but they have much bigger leaves and don't spread so aggressively.

You have seen what it does, now you have to decide if what it does is what you want there.

Expect it to get taller and denser if leaf in place.

In late spring the living oxalis would not have been spitting seeds yet, your new plants are from dormant seed and disturbed soil. You will always have oxalis in this bed to some degree.

I would keep it in control by pulling, especially if it is flowering. The seeds will propel easily several feet, perhaps to adjacent planting areas. Use a little tool to poke in the ground by each sizable plant so that you get the surprisingly long tap root with the plant.

It's a weed, aka 'Japanese clover'. Commonly found in nursery stock when the potting soil is recycled. It's an Oxalis and loves disturbed ground. Pull it before it goes to seed, which it is probably doing already. As mentioed above, get all the root. It tends to break off just below the soil level when pulled. A mulch as thick as you can get away with will help in the control of it in the future.

This looks to me like the type of oxalis that grows from little bulblets. Likely were spread about from the original clumps you had before digging the bamboo.
You may be able to DIG, lift and remove some of the bulb clumps to help control the amount you have, but will likely leave many little ones behind. (Pulling just leaves the bulbs behind, but may deprive them of some nutrients enough to slow them down.)
What color flower do yours have? Pink?

I'd be surprised if the flowers were any other color but yellow.

Oxalis debilis?

If I have this plant should I be worried about it becoming a weed in our climate & soils?

Ours looks just like these with the light green leaves and pink blooms. http://www.barnstapleallotments.co.uk/weeds.htm