Can someone able to clarify these statements at all, as they seem to contradict one-another.
In truth, no they don't, but it is complex, I grant you!

Perhaps the second paragraph would be better-served if the word 'sit' was replaced with 'stay'... which is what it's explaining, really. If you
stay at the 60% you've reached in order get into a war/election with the ruling station's faction, you will not get the other stations... you will almost certainly have to move influence levels again, to match them and fight with them, too.
When we put our minor fraction into the game we chose the largest station assuming we would not have control of the system if we chose one of smaller ones. This apparently is not the case. The issue is the other two big stations in the system are far superior trading stations and so bring in large amounts of money for the other large fraction (the previous system owners).
So we are trying to get our influence over 60% to try and take one of these two stations. But that second quote above implies this will not happen?
The 'get over 60%' rule only works for causing a conflict with the ruling station's faction. In order to conflict with any other station's faction, you have to match them, or nearly so.
Also does the station you take, and hand in a mission in have an effect? Right now we are doing missions for our fractions from one of the stations we need to take over, hoping it had it's own hidden influence.
Complex answer, potentially. Broadly speaking, doing missions for Faction X, regardless of which station or outpost or even planetary base you're handing that mission in or doing it for, primarily affects Faction X's influence, not the station! There are subtle variations to this (most likely when the mission involves an action that also has an effect on the station-controlling faction's influence, such as murder, smuggling, etc) but basically, missions improve (or damage, if failed/abandoned/flipped - to varying degrees) the mission-giving faction's influence.
However, this might not be what you want! It may be prudent to actually harm your own faction's influence, in some cases. For example... your faction, Faction X, has 66% influence and controls the major station already, let's say. But on the rise to power, you missed a potential fight with another faction, Faction Z, and thus don't control their station, which is still having a significant effect on trade in the system. You want to get control of that secondary station, and Faction Z's influence is currently 34% - in order to fight Faction Z and possibly win the station, you will have to either increase Faction Z's influence or decrease your own Faction X's influence - or quite possibly BOTH those things, until you and Faction Z end up in a battle or election - probably around the 50% mark, in this case (halfway between both factions' original starting point). In practice, it probably won't happen spot on 50% - it'll be somewhere within a few percent of that area.
Bear in mind that negatively influencing your own faction's influence downwards, can be tough (you don't generally want to end up wanted in your own system - it can complicate matters immensely, for example). Sometimes it's better to focus (and have your team focus) on positively affecting the rival faction (Faction Z) and letting your own faction's influence 'drift down' under its own steam. You'll need to ensure none of your team is positively affecting your stations: no trading (in or out), no missions for your faction
anywhere in that system. Eventually, Faction Z will start coming up, Faction X will start to go down, and when they meet in the middle, be ready for either an election or a war/civil war, and be prepared to fight to win it! Election fights are via trade and non-combat missions only (boost your team), war/civil war fights are through combat in a combat zone (fight for your side) and possibly through combat-related missions for your faction (though I am unconvinced about that, personally, but only because I've never encountered any in that situation yet - they may well exist).
Hope this helps!