False. There were viruses in the PC and Mac world for years, long before the 1988 Morris worm incident popularized the "worm" term.
Viruses spread via interactions like booting an infected floppy disc, or running an infected program copied from another user.
A virus is simply a piece of malicious code which attaches itself to programs, arranges for itself to be executed when those programs to be run, and thereby spreads to more programs as programs are copied from system to system by unsuspecting users.
A Trojan horse is a malicious program which a user is somehow fooled into trusting, installing and running. It doesn't have to be a virus at all; for instance, it could be a fake authentication dialog that steals their credentials and then defers to the real authentication.
A malicious thumb drive deliberately dropped in the parking lot of a company is a modern example of a Trojan horse. It might not infect anything, just steal information and transmit it.