Hi Jason, welcome to Outdoorking. First, the symptoms you described for your engine are typical of overheating, especially if due to low oil level. So far as your 300 ml figure for how much oil it needed is concerned, are you aware that Honda screw-in dipsticks, unlike Briggs screw-in dipsticks, are required to be read with the thread completely unscrewed? If you weren't aware of that, your 300 ml estimate for how much oil it needed will be a long way short of reality. Incidentally what you have done has caused more damage to your engine than several years of normal use.
Briggs engine numbers include the year of manufacture, but as far as I currently know, Honda ones do not. If you post the full serial number including prefixes etc for your engine, I will see what I can find out about it, but it probably won't include a date. If you post the actual model name and serial number of your Cox mower, someone may be able to estimate the date from that.
