Ahh...
ok, haven't read what Inara says, but
- most of the Engineers will accept one or another additional thing to advance in standing/rank with them. For Felicity, it's exploration data
- however, the fastest (and easiest) way to rank up with any engineer is to have them do mods on your ship/equipment. Usually, you can just mod one piece of kit up from nothing to rank 5 without losing anything. Keep in mind that, unless you have reached rank 5, modding will cost you more attempts
- this only works with engineering on their base, though. Remote workshop upgrades don't count
- the other thing is those mystical "10 million in exploration data". That's the usual rough figure (give or take -50/+100%) for getting allied with any (minor) faction in the game. Drop into any starport, sell ~10 MCr. worth of exploration data, and a) you will become allied with the station owner and b) said station owner will rceive a big boost for their in-system influence (INF). This suffers from diminuishing returns - optimum for BGS operation is more like 1 MCr.
The exploration data would be helpful for Engineers like Marco, who requires a certain amount of coziness with Sirius... At least I think it was Marco...
In case of Elvira or Tod, I don't have the feeling that the relation to the resident factions does make a difference. Tod was the first engineer I visited after leaving the Pilots Federation, having collected the amount of bounty in the security of their little space. I didn't do anything in the radius of influence of those factions resident to the system Tod is based in. But he still welcomed me and gave me all the upgrades I asked for, eventually reaching level 5 access.
On the other hand, in Suhte, which I consider my temporary home base, I did a lot of paid work, mainly exploration and data hauling (and the odd bounty for an upcoming pirate / mercenary that thought it would be a good idea to interrupt my work). I am allied to Canonn (the main faction around there) and somewhere around three other factions of that region. But that didn't change anything about the accessibility of Long Sight Base and it's inhabitants.
I cannot say much about Felicity, though, as I just recently unlocked her. But I already have tier 5 acces on her base...
Besides all that, I have not seen any hint on this mechanic in game.
Regarding EDDB/Inara info (they both draw their data form the same database): this info is added and updated by the players. To be exact, by those players who have one of the feeder apps running on their PC while playing the game. Whenever you dock at a station, that app grabs the commodity prices, outfitting and shipyard info from your journal and sends it to the database. Downside is that everyone else then can immediately see that data, so if you've discovered a valuable trade route you want to milk for yourself, you might want to set your feeder app into offline or delayed update mode. Of course, the station's inventory and prices change over time and with the BGS, so if nobody has visited there for some time (depending on circumstances, that may be as little as two hours), the info you can see on EDDB or Inara is basically worthless. Whcih is why they usually also show the time since the last update.
I take it this feeder app grabs
all available data from that station, no matter what ship the player is in or which tabs they look at.
Anything else would make this system utterly useless.
In a way, EDDB and INARA are relatively useless for me, as I do not have the computational power to run two screens - one for the game, the other for my browser. And real life does not mesh very well with this, either...