That's your interpretation, fair enough, but I don't think it's entirely accurate in terms of the pre 0630 situation.
Take this as an example.
Southport - Chester; changing at Liverpool Central.
It is perfectly valid with an all zones SaveAway or a Merseyside 60+ concessionary pass to board the 0622 Southport - Hunts Cross service, alight at Liverpool Central (0712), and transfer to the Wirral Line platform for the 0715 service to Chester. However, if challenged, it is probably sensible to take with you a printed itinerary that shows your journey commenced prior to 0630, because it would NOT BE VALID to board the same 0622 ex Southport service at, say, Ainsdale at 0631, or any later station. As Ainsdale is unbarriered, it is highly unlikely that one would be challenged at that point, but that doesn't alter the fact that such a traveller's SaveAway or a Merseyside 60+ concessionary pass is NOT valid for this journey, whereas the identical ticket for their co-traveller who boarded at Southport (or Birkdale, or Hillside) IS valid.
Regardless of whether the barrier staff at Southport are in the habit of permitting all zones SaveAway or a Merseyside 60+ concessionary passholders to board the 0927 departure (technically this is NOT permitted) then as
@JBuchananGB states, it is fine to board this service at Birkdale (0931) or any later station.
(As an aside, this problem "goes away" during the imminent 'Autumn Timetable' starting next Monday, when the erstwhile 0927 ex Southport is retimed to 0924, and thus leaves Birkdale at 0928 - so still too early for passholders!)
An interesting issue would arise if a traveller's intention was to travel from Southport to (say) St Helens Central. As discussed above, if he/she boards the 0622 departure at Southport, this journey is fine. Suppose this traveller goes to Liverpool Central (0712), leaves the station, and walks to Lime Street to catch the 0739 (Blackpool North) service. Is this a new journey? The traveller has left the railway premises, so it could be argued "Yes". But if he/she alighted from the 0622 Southport - Hunts Cross at Moorfields (0710), and transferred to the Wirral Line for, say, the 0716 ex New Brighton to make the two minute trip to Lime Street, and then went up in the lift to the main concourse at Lime Street, it could be argued "No - it's a continuation of the same single journey".
Potentially a nightmare for enforcement!