Does "rent out" only refer to when the owner offers something for others to pay to use, or also to when the user pays the owner to use something?

This is probably not correct usage; a verification in an online up to date¹ dictionary shows that "out" is used only in the case when "rent" means "receiving" money in return for allowing someone to use your property, and at that it is an optional particle.

¹ _{The dictionary used, from which is taken the following reference, results from a new release in March 21}

(OALD)
1 [transitive, intransitive] to regularly pay money to somebody so that you can use something that they own, such as a house, a room, some land, etc.
♦Are you looking to buy or rent?
rent something to rent a house/an apartment to live in rented accommodation/housing/property
♦ The property is available to rent on a short-term lease.
♦ We're looking for a house to rent in the area.
rent something from somebody
♦ Who do you rent the land from?

2 [transitive] to allow somebody to use something that you own such as a house or some land in exchange for regular payments
rent something (out)
♦ We rented our house out for a year when we went abroad.
♦ She bought a three-bedroom flat with the objective of renting two rooms.
♦ He is making a profit from renting out the property.
rent something (out) to somebody
♦ He rents rooms in his house to students.
♦ The land is rented out to other farmers.
♦ She agreed to rent the room to me.

I rather believe that you can trust your instinct and that the somewhat confusing word context is more likely to be at the root of an error in that speaker's English. An additional fact that tends to confirm this use of "out" is what can be read at the entry for this adverb in The SOED.

(SOED) 2 Expr. motion or direction away from oneself, a center, a familiar place, the shore, etc., esp. to a remote point, to sea, to war, etc. Now also spec. (of a boat, train, post, etc.) going out, departing. [since] Old English b From one's control or possession into that of others [since] Late Middle English.

in the same entry is still found the following.

6 To an end, completely, thoroughly, fully, to the utmost degree. Middle English.

Nevertheless, "2b" seems to prevail and coincide nicely with the sense in n° 2 of OALD.

Remark: It is to be noted that this verb is not considered to be a phrasal verb, at least not in OALD.