While lifting my right lat got strained and now it hurts to move around and breathe in deeply. is this more likely a tear or a pull?

Tear vs pull no diff: Most of the time, those words mean the same thing, an injury to a muscle which will heal. Based on your description that what it sounds like, a muscle injury that will heal. If the tendon were ruptured, surgery is a consideration. If the lat tendon ruptured, there would be lots of bruising & deformity in the armpit. This is unlikely. The most common risk factor for a lat tendon rupture is steroids.

See below: Hard to tell although a muscle strain or pull is likely a partial tearing of the muscle fibers. This can be small and microscopic level or be large and obvious on mri. It is not unsual for there to be trouble taking a deep breath with this. If it is not improving after a week or so, you should see a doctor and get checked out. Sometimes physical therapy can be helpful for this.