Well, I investigated, and I was wrong. BBSes did predate the internet—but not, as you say, by several years. Rather, the time gap was about six to ten months, because the internet (as distinct from ARPANET) was created in late 01978, not the early 01980s. (Also, we didn't use acoustic couplers
instead of modems; modems were what we were coupling with our acoustic couplers.)
Ward Christensen and Randy Suess put the first BBS, CBBS/Chicago, online in February 01978.
As for internets, Louis Pouzin proposed internetworking in 01974, and Cerf and Kahn published "A Proposal for Packet Network Intercommunication" https://ieeexplore.ieee.org/document/1092259 the same year. Within the ARPANET project, the Internet Experiment Note series began in 01977. IEN 1 https://www.rfc-editor.org/ien/ien1.pdf is dated "29 July 1977". But, although it's talking about "the last couple of years" and "the ARPANET internetworking community", it seems to be talking about proposals for networking protocols to implement, not reporting results from actual experiments. IEN 65 https://www.rfc-editor.org/ien/ien65.pdf are the meeting notes from the TCP meeting of August 5, 01977, including an assignment of what would later be called "class A" IP network numbers; for example, network 18[.0.0.0] is assigned to LCS at MIT, an assignment MIT still retains today, and network 10[.0.0.0] is assigned to ARPANET, an assignment it would retain until it was shut down. But that doesn't mean they could actually send packets with those addresses yet. At that point they were still considering things like variable-length addresses (in IEN 66).
Even in IEN 22 in February 01978 https://www.rfc-editor.org/ien/ien22.pdf they are talking about all plans to set up routers in the future tense, while IEN 46 from June 01978 https://www.rfc-editor.org/ien/ien46.txt talks about MIT already having two local networks (apparently participating in the internet experiment) and concerns about "upheaval to (...) gateway [router] code".
IEN 51 from July 01978 reports high levels of packet loss in the SATNET gateways https://www.rfc-editor.org/ien/ien51.txt, suggesting that UCL was somewhat successfully internetworking at that point.
But IEN 53 from August 01978 opens saying, "Vint put the stress on the need for the Internet to be a working system very soon," proposing various milestone dates for 01979, though it also reports that "3 gateways are up between SATNET & ARPANET", and that an internet was up and working at PARC interconnecting 22 to 25 ethernets over PRNET, but presumably not using IP (at the time called IN).
IEN 60 from October 01978 https://www.rfc-editor.org/ien/ien60.pdf section V reports, "Testing of this [new shortest path] routing algorithm [is] in progress[,] and it should be operational in the ARPANET/PRNET gateways and the ARPANET/SATNET gateways by the end of this year."
Later that month, IEN 63 https://www.rfc-editor.org/ien/ien63.pdf reports, "A number of [Internet] feasibility demos have been done. We need to show an operational [Internet] capability. In June 1979, eighty users will be online via PRNET in Ft. Bragg. In April 1979, there will be a PRNET demo at Ft. Sill. In May-June 1979, UCL will be disconnected from the rest of the ARPANET and will depend on the Internet system." It also reports that at BBN the SATNET-ARPANET router and the PRNET-ARPANET router are now operating, and asks, "Is IN [IP] available directly without TCP?" Forgie at Lincoln Lab reports, "Hope to have an internet speech capability up by the end of the year." This is also when today's minimum MTU of 576 octets was established.
In IEN 76 https://www.rfc-editor.org/ien/ien76.pdf in January 01979, "Ginny [Strazisar at BBN] noted that both the SATNET and PRNET gateways will run both IN-4 [IPv4] and old protocols by the end of January." So at this point they did in fact have the internet up and working. BBN reported success running TCP/IP ("TCP-4") on PDP-11 Unix, Noel Chiappa at MIT LCS had four running internet nodes on Multics, and at NDRE in Norway "TCP-4 has been running for about three months".
So I think that, by any reasonable definition, the internet that we're using today was up and running in late 01978, on multiple operating systems and multiple continents. It just wasn't very big yet. The packets they were sending would probably have been interpretable by today's Wireshark, though I'm not sure about that (IEN 54 defined the IP header format that was standardized in RFC 760, but there might be subtle incompatibilities, and I'm less sure about TCP), and even some of today's IP-address space allocations were already established. If you were to bring up a software emulation of the Multics TCP/IP stack on your LAN, you could probably telnet to it.
The other internets like the one at Xerox PARC might have predated the TCP/IP internet we use today, but not by more than a few months—not by enough to predate the BBS. IBM's internal corporate worldwide computer network was a few years earlier; I forget what it was called, but I don't think it was an internet.
My error was that I had thought that there were lots of internetworking experiments in the years leading up to IPv4, given that the concept was published four years earlier. I didn't appreciate the slowness of the development of the necessary software and the resulting degree of preplanning and deliberation. BBN's Unix TCP/IP was written in PDP-11 assembly, and presumably the TCP/IP stacks for Multics and the PDP-10 were also written in assembly, which may be one reason for the slowness.
As for "what most people today think of as the internet", ignorant people have all kinds of stupid misconceptions. They think that cellphones send radio signals to satellites, that Christopher Columbus discovered the United States, that Henry Ford invented the automobile, that many people eat too much salt, that microwave ovens are radioactive, that vaccines cause autism, and that Xbox Live and WhatsApp don't use the internet. But presumably nobody that ignorant is participating in this discussion, so I don't know why you'd bring it up.