Giving a precise answer to this is I suspect impossible, as the very notion of physical existence is quite subjective. Please therefore treat this answer as subjective - I would not expect all physicists to agree with it. But here goes…
As a first stab, I’d be inclined to reason a bit like this: Your suggested philosophical definition of ‘having extension in space’ does correspond to human instinct about what existence means, so let’s see if we can refine it to match physics:
- You need to say, ‘spacetime’ not space, since time is – so far as we can tell - not separate from space.
- It may be better to say something like forms part of spacetime rather than ‘in’ spacetime. That’s because most physicists would view spacetime itself as having physical existence.
Now you have a definition something like ‘X physically exists if X forms a part of spacetime’. Which is quite good, because all the physical objects around us (like electrons, and photons, and tables, and snazzy new hi-fis) can be regarded as perturbations of spacetime, and therefore being part of space-time rather than merely existing in it. So this definition does match intuition quite well.
There is something more though. Physics is ultimately only interested in whether something causes measurable effects – because that is after all the only thing we are capable of discovering about the Universe we live in. Suppose something existed in the Universe – say 2.5m x 1.5 m x 0.5m in size - that had no ability to affect us or any of the objects we can measure in any manner whatsoever (the ‘invisible pink unicorn’ ;-) ). Such an object would be completely undetectable to us and therefore of no interest to physics. Indeed, a scientist would probably argue that my words in the previous sentence ‘suppose something existed...’ are nonsensical, since, if there’s no way for us to detect it, then as far as we are concerned, it is completely imaginary, and therefore cannot be seen as existing in any real sense. I suspect that is where physics would differ from philosophy: Physics would not consider such an object to have any physical existence, whereas philosophy would consider the theoretical possibility of that object existing (after all, I’ve just given its dimension, so it does have extension in space!)
Because of that reasoning, I’d be inclined to add an additional criteria to a physicists definition of physical existence: X physically exists if X is part of spacetime AND is able (either directly or indirectly) to influence other objects that are a part of spacetime and are detectable to our senses.
I still don’t think that definition is quite complete, although it may well be as close as you can reasonably get. The main issue with it is it doesn’t account for things like ‘logic’ or ‘ideas’ or ‘beauty’ – clearly, all those things do influence us, and therefore are detectable (albeit subjective), but most people would say they do not exist in any physical sense. I’m not sure off the top of my head what precisely makes those things ‘not physical’, and therefore what else might need to be added to my definition.