Are you sure its HDMI? Or otherwise are you sure its DV? The 2 formats are mutually exclusive. DV was an SD interlaced and typically 4:3 format that used FireWire as its proprietary transfer connection. Never HDMI to the best of my knowledge. HD in HDMI stands for high definition, so obviously it replaced firewire when HD cams started to become available. It sounds like you have a HD cam (not DV). There may be a setting on the Canon menu that selects or switches on the HDMI output. But like you say HDMI seems to work with your TV. There must be a bit of software that controls the Intensity card. Maybe you need to select in input setting or format.
I'm looking at adding an Intesity card so I'd be curious to see how you get on and if you ever figure it out. I've heard BM cards can be pretty finicky to work with.
I KNOW this was from a year ago but I feel bad for anyone new trying to figure this out. OK so DV is Digital Video but so is any video these days, that version was just stored on a tape. I have DV cameras still in use today with a 1080i output (1920 x 1080 Full HD interlaced)
The reason a 480i or old VCR would appear is because the capture device is set to capture 480i signal (set in the capture settings or "Properties" in OBS) Connecting almost ANY device to a TV will likely work because the TV will automatically match the incoming signal.
TV's are designed under the assumption that the user may not know what they are doing, so the TV does the work to set things properly. When buying a professional level device like " BlackMagic Intensity Pro 4k card " it is assumed that the user DOES know what they are doing and will change the settings accordingly...
Setting your Blackmagic device (or any other device) to capture 1080i @ 59.94Hz or 60Hz will suddenly show the HDMI image. The age or type of device is basically irrelevant as long as you set the capture to match the video signal type. If you are unsure, your TV may provide some insight when you connect the device or camera. If the TV input shows 720P 60Hz, that is probably what the camera or playback device is sending. If the TV displays "HDMI Input 1 1080i", then you know what to set your capture card to. You may have to try 59.94 or 60 etc. but eventually, one will be correct.
Due to the fact that HDTV standards are 480, 720, 1080 and now 2160 resolutions @ 23.94, 24, 25, 29.97, 30, 50, 59.94 AND 60Hz, there are many options to choose from. You need to set the device to match the capture OR set the capture to match the device but it is unlikely you will ever see an image until these two are both correct. This is just the nature of Digital Video resolutions, frequencies or frame rates.