Usually a routine stool test (O&P, or C&S) does not include H.pylori.
H.pylori does not live in the 'stool' containing guts, is not an intestine pathogen culprit, hence cannot be cultured from stool, BUT its dead antigen (body parts if you wish) is still excreted, are not gone by the time getting out with stool, even being dead.
So, yes, H.pylori can be very well tested in stool via a certain antigen test,
but no, it is not in the routine panel of stool pathology
(Please get a printed copy of your stool result and have a look)
As for all antigen and antibody tests in medicine, there is hardly anything 100% in specifity and sensitivity. That is really asked much.
The H.pylori testing, be it stool, breath, biopsies, blood are really pretty good tests compared to many other pathogens, but not 100% still.
(breath and stool tests coming pretty close to it if no PPI was taken 2 weeks prior.)
Having taken PPI doesn't make the test 'gaga', but it lowers the sensitivity, e.g. biopsies (sample of tissue during endoscopy) for quick urease test are lowered from ca 92% sensitivity to 75% under PPI, depends what study you read about re exact numbers, but it is lower so or so. You still get correct positives, but not as many.
The blood H.pylori test as a huge disadvantage:
it doesn't say, if the infection was going on right now or passed since the antibody, the immune response is tested only. (in H.pylori only IgG is sensitive enough in blood to be tested for ... IgM would say more about current infection, but the test is so bad in sensitivity, that it must be abandoned).
If H.pylori IgG in blood was positive, you have or have had or had a H.pylori infection. If negative you might be in the diagnostic window and false neg just developing H.pylori IgG or be really negative. So....blood test is good, but one needs to know how to interpret it and when to use it.
The breath test and biopsy look at urease activity of H.pylori, a metabolite of h.pylori, so only live bacteria can do this.
Biopsies also look via staining of the physical presence of bacteria. Of course that can go undetected by a human eye (they are smaller than a nuclei of a cell and if only one in once microscopic slice, it is impossible to be detected) or be taken from a part in stomach, where no H.pylori was living in the first place.
It's a random sample.
In your case, I would look at the stool test result and if no H.pylori was tested for,
I would ask for a Carbon urea breath test (CUBT) and all shall be pretty clear.
Best wishes!