The short answer is you’re close with your first guess - by squeezing the bottle a drop forms at the tip that is stable due to surface tension, and then steering that drop into an open eye (usually with the help of a partner doing the actual instilling). This means the bottle tip itself never touches the eye if you’re doing it right, which is good as these bottles are reused throughout the mission (though they are crewmember specific, so you’re not using someone else’s eye drops).
Most of the eye drop use is related to SANS (Spaceflight Associated Neuro-ocular Syndrome) imagery or data collection, which occurs periodically during the mission. While we no longer require fundoscopy, pupil dilation aids in image acquisition for other modalities, and anesthetic drops for tonometry are a must.
CSA’s Hadfield posted a good description of how the entire tonometry process works, including drop instillation, here: