I too agree that an exercise performed while standing (stair climber) will result in a higher HR because more muscles are being activated. Mark definitely had some very good points as did all the others. Karin also had a good point that the 220 minus the age is just an average across the board and does not take into account the clients physical health. Another thing you might want to use is the Karvonon formula which takes into account a persons maximum heart rate and their resting heart rate.
To find ones resting heart rate (RHR) they would need to take their pulse for 1 minute at least 3 days (preferably 5 days) in a row after waking but before rising out of bed without an alarm waking the person up. Write these numbers down then add them and divide by 3 or 5 depending on how many days they took their RHR. This will give you the persons average resting heart rate (RHR).
220 minus age = maximum heart rate (MHR)
MHR minus RHR = HRR (heart rate reserve)
HRR multiplied by training % + RHR = bpm
Ex: 50 year old with a resting heart rate of 65 bpm who wants to train at 70%
220 – 50 = 170 (MHR)
170 – 65 (RHR) = 105 bpm (heart rate reserve)
105 X .70 + 65 = 139 bpm
As someone else already pointed out, use your best judgment. You know your client better than I do. If it is unsafe for your client don’t have them train that hard. Using the RPE may be another option.
Hope all of this helps!