I doubt you had fibre in 1968 since it was not created by Corning till 1970 and initial trials in 1977
http://www.greatachievements.org/?id=3706
Most likely was an early co-ax cable TV service, at a guess Rediffusion
Fawley is marked as coming soon for Openreach fibre, and that would be run to the street cabinet that your line is served from. If you have no street cabinet then you may be a Exchange Only line, which is NOT compatible with the two fibre rollouts Openreach is doing currently.
As others have suggested your postcode into the various checkers, or even better the telephone number may reveal more information. For interpretations copy/paste the text and people can explain it.