It's underlined in Section 11.19 in the CBA.
Basically any player signing an NHL contract from a Euro League (Liiga, SHL, KHL, NLA, DEL, Czech League) or any of their lower level leagues has the option of including a Euro out clause in the first year of their contract. It's really rare for anyone outside of ELC's to exercise it though. Both Soderstrom and Kravstov used their option to include one in their contracts and they came out of the SHL. Vesalainen had one in his contract and used it to go to the KHL, not even back to Liiga.
It really has nothing to do with Liiga, it is a players right to include one in the first year of their contract if they are coming from any league in Europe. Even if they played in the NHL, went to Europe and then signed into the NHL again they would have the right to include a Euro out clause in the first year and exercise it at their discretion. (an example of this would be if Jesse Puljujarvi signs with Edmonton again. He would have the right to include a Euro Out Clause in the first year of his new contract since he signed a new contract in Europe) Very few older players do as it likely raises some suspicion from the team signing them. But it seems younger players are exercising this right a lot more.
Igor Shesterkin - CapFriendly - NHL Salary Caps
Another example is Igor Shesterkin who also signed his ELC with NYR in the offseason. He is coming out of the KHL, but optioned to have a Euro Out Clause in the first year of his ELC.
The Euro Out Clause further states that the player has the right to sign with any team in Europe. Not just the one they came from, or even in the same league they came from (Like in Vesalainen's case). They also don't need to have a Euro contract in place, they can test Euro free agency and sign with any team that wants them in Europe, in any Euro league.
NHL teams have zero say in the matter. If the player wants one in their contract, they get it, and if they choose to exercise it at any time the NHL team must comply. The only sitpulation is that they must be signing out of a European league. So if a player is signing out of a CHL league or the USDP or the AHL. They cannot include one, even if they were born in Europe. Hence why under this rule, Auston Matthews could have included one in his ELC, since he was being signed out of the NLA.