Yep, that's it. I just tried it with a file and it works flawlessly. The container (.mov or .AVI) does not really matter as long as the codec can be read by the client. That's we I advise encoding to MP4 as this is one of the most compatible formats.
Ronnys trick also works well for changing .MOV files to .mp4 in cases where certain Windows PC's cannot playback MOV files. Years ago we did this all the time for clients, but nowadays more PC's deal better with MOV files, typically those that have iTunes installed.
OK, the client is mistaken. Someone working with them is very PC illiterate and thinks AVI is still some sort of standard. Microsoft dropped support for it ages ago. Just give them a normal H.264 (.mp4) file and ask them to try it out. Explain that AVI is very old format that Microsoft doesn't support any longer. Ask them to try out what you're giving them.
If they THEN want AVI for some reason, do what Ronny says and be done with it. But I'd at least try to educate the client first. In my professional experience, they love you for it.
AVI files are just a container -- various different codecs could be inside. What Karsten said is correct: sometimes non-video people use it to mean "video file".
For a public performance, I would not give them *only* an AVI file (real or renamed). Along with the requested AVI, I'd give them H264 MP4 files at 720p and 1080p using the default FCPX Master File>Computer>H264 output. Some old computers will handle 720 better than 1080.
IF they demand actual AVI, you could export ProRes, get the free version of Resolve, and export one of the various AVI codecs it supports. Plus give them MP4 at 720 and 1080.
I've seen many playback failures due to situations like this. The equipment is often old or of unknown configuration. More recently it happens if giving them 4k "because it's better", then the playback system can't handle it or stutters. Lots of older projectors can't even do 1080, or are 4:3 aspect ratio.
Audio is another issue, esp if using a separate audio chain to the PA with video driving modern large-screen TVs. The TVs often have video processing enabled which can introduce lip sync issues.
If it's a hand-off situation, you can only give them several formats, some advice and hope for the best. If it's an important showing where you'll be present, the best solution is get access to the venue early and personally test the equipment, also if possible bring back-up laptop and various cables and adapters.
My doc team finally gave up on venue assurances their equipment is fine - we take our own PA, projector, computers and screen.