The problem with Edinburgh is despite having a taxi rank, it's not outside the terminal (that area is reserved for buses). Since you are presumably arriving from Sweden, it will be a longer walk than from the domestic terminal. It's not a ponderous distance, but something to consider if you are carrying lots of luggage because getting to the taxi rank is something you have to do unassisted.
Private car service versus taxi
Taxis are normally a saloon car which will be fine for two people + 4 suitcases. The taxi fare will generally be cheaper. The downside is that cabbies assemble at the rank when they know there will be lots of arrivals, so if your plane is delayed or you spend a long time claiming your luggage, the rank may be empty and you will need to join a waiting queue. I got an indicative estimate from a black cab service of GBP 35 Edinburgh to Linlithgow city centre.
The private car service will have the driver holding a courtesy placard with your name on it (normally at the 'meet and greet' point). He may help with the luggage.
Contributor Patricia Shanahan has commented...
On delay causing a private car service to abandon a commission: I
don't have experience at Edinburgh, but I have used private car
service at several other airports. In each case, the service asked for
the flight number and monitored the live flight information. They
routinely adjusted their pickup time based on that, accommodating even
I got an indicative estimate of GBP 45 for a private car taking you from Edinburgh to Linlithgow city centre.
In both cases you can book by phone, or email, or an app. Almost all the major private cab and regulated cab services in the UK have downloadable apps for mobiles and iPads; and a few even have a toll free number.
There are also unregulated drivers who may approach you at the terminal exit. Pricing is done by haggling with them, but they will invariably be cheaper than a regulated driver. 'Caveat emptor' applies.
For your question about buses, buses leaving from the airport are fitted with a purpose built luggage area. I do not see a problem for two people + 4 suitcases unless somebody gets on before you with a half-dozen steamer trunks. I got an indicative estimate of GBP 6 Edinburgh to Linlithgow High Street.
There are also helicopter services out of Edinburgh Airport, but you have to walk to their mooring zone, and it's farther than walking to the taxi rank. It's the quickest way (and the priciest way), but you will need a local taxi from their drop off point in Linlithgow.