The meters show the test ON THE METER. You will absolutely see the results. The CoaguChek XS shows the INR - you have to push a button to see the prothrombin time (if that matters). The Coag-Sense shows both.
I've been recording the prothrombin times for every test I've had (if the lab provided it) for the last decade, but I'm not entirely sure why, because, depending on the reagent used, the time may vary somewhat.
Your cardiologist or PCP will be the one prescribing your warfarin -- I suggest that you create a spreadsheet with time, day of test, (in my case, the meter used for the test), if it's a lab test, indicate this, your dosage and anything that may have impacted the test (splurged on salads, took an antibiotic, etc.). If you show THAT to your doctor, indicating that you've got some control of your INR and dosing, your doctor may have little hesitation before prescribing your warfarin.
Your cardiologist probably won't have much interest in adjusting your dosing - he or she may pass it off to someone in the office who is concerned about those things, or send you to a clinic or service. With the resources here (Pellicle is a great resource), you'll be able to properly self-manage your anticoagulation. You'll only need your doctor for prescriptions and prescribing occasional blood draws (if you're interested in comparing the results to a lab - which may get it wrong).
(My PCP trusts me. He knows that I self-test and self-manage, and freely prescribes my warfarin. He also understands that it may take a few different doses to create my appropriate dose. Currently, I get 7.5 mg tablets, 4 mg tablets, and 1 mg tablets. I could probably do as well with 5 mg tablets as I do with 7.5 - but my INR frequently requires a dose of 7.5 mg, so I only have to take one pill instead of two. My doctor had no issues with prescribing multiple doses. You may have to educate your cardiologist or PCP about the reasons for multiple doses, but this gives you the flexibility to create a more consistent daily dose than being stuck with one dose (if a smaller or higher dose would work better) for you).