Can I have a lip lift despite the fact my teeth show whilst my mouth is relaxed and open? (Photos)

Based on your photos and the measurement you shared I would agree, you seem to be a great candidate for a lip lift. If you are successfully measuring the distance between the bottom of your nose and the top of the vermillion border of your upper lip as 2.1 cm, a lip lift would likely achieve your goals.
When the lip is thin and the distance between the nose and lip is greater than 1.5 cm, fillers can cause a duck-lip affect, a lip lift and fillers typically produces a much more natural looking, fuller result.
Be sure to consult with board certified plastic surgeons, it's absolutely okay to get a 2nd and 3rd opinion.
Best wishes!

It appears in the photos and by your measurement that you are a candidate for an upper lip lift. Patients should show between 2-4 mm of upper incisor at rest. When smiling, the entire incisor who can be exposed along with 1-3 mm of gum tissue. I think that you would have an excellent result with an upper lip lift surgery. 

Hello,I do as many as 5 of these a week (lip lifts), and as far as I can tell from your photographs you are a perfect candidate.  You do not have much tooth show at rest and should be quite happy with the procedure.  Sometimes we further enhance the lip with a dermis graft at the same time which can add some additional permanent volume.

There are surgical and non-surgical approaches to lip augmentation. Non-surgical approaches include botox (slight lift, reducing gum show) and filler (volume restoration). Surgically, you can have two separate incisions under the nose (italian), one incision under the nose (bull horn), one incision at the red/white upper lip junction (gull wing) or two separate incision at the corners, or a combination, depending on your desired look and anatomy.
In your situation, you are a good candidate with the length of your philtrum and thin lips. We need to also keep in mind your bottom lip and making sure it stays in proportion to the upper lip after a lift. Additionally, there are dental procedures that can address the "toothy" look, so I would not let that deter you. 
Consult with an experienced lip surgeon for a complete range of options that are appropriate for your situation.
Best regards,
Gary Linkov, MD