You should assume the Home Office has your exit dates. By 2014, most outbound routes were covered for API data. This data is retained for 10 years. The question you should be asking is whether or not the decision maker has access to this information.
If your student visa was curtailed at anytime, you should assume you have been marked as having been “removed.” This is because the Home Office runs a program whereby they search for students whose leaves have been curtailed against API data to find students who have not left. Students who leave after their leave by date are deemed to have been removed.
If you properly finished your course, and did not have your leave curtailed, chances are the Home Office still doesn’t “know” about your overstay yet even though they have the data.
However, the ECO may request a search of the outbound travel records to determine the date of your departure. This could be done, for example, if the ECO can’t collaborate travel dates from your passport. Again, records more than five years old are restricted so even if the ECO requested this data, chances are he might not know. Since April 2015, overstayers (leaving after that date) should automatically be tagged as such in the ISA system for all decision makers to see. That system still has limitations.
The chances of a visitor visa after a 2 year overstay in the last 10 years are close to zero. If they find out you lie, the chances will be even less.
To sum up: the Home Office had no systems in place to automatically detect overstays in 2014. The only way they would know is if there has been human intervention at some point (this could be a new application where they manually checked your travel records). Since your outbound record is still in their database, it would be risky to ask for a SAR. This is because, if they were to discover your overstay, your record will be permanently marked. The safest route is waiting until 2024 when your outbound record is deleted and then making a SAR to know what data they hold.