What device to use to chop/dice tomatoes

Cut the tomato in half at the midsection or equator (stem end being the "north pole") to expose the seed cavities. Holding the cut side down over your garbage bowl or trash can, gently squeeze to remove the seeds. You can easily pry out stubborn seeds with your fingertips.

Place the tomato half cut-side up on the cutting board (Cutting waxy skinned vegetables is much easier when the skin is on the bottom so the force of the knive cuts cleanly through the skin with less effot) and cut in half again. After doing this with both pieces the tomato will be in quarters.

If you want evenly diced pieces of just the exterior flesh (such as for sprinkling on a plate for a garnish), then use your knife to cut out the interior portion so that just the tomato "shell" remains. You can then easily dice the exterior shell.

Otherwise, after cutting into quarters just gently push down to flatten each quarter and cut into strips lengthwise and then cut cross-wise to finish the process.

While cumbersome to explain in text, it's much safer and quicker than cutting the tomato into slices and then trying to dice from there.

Tomato puree is concentrated tomato. So where it is called for in conjunction with other tomato products, be they fresh or tinned, it is an attempt to increase the tomato flavour/presence in the dish without increasing the bulk or water inherent in less concentrated products.

Additionally the flavour of puree, having gone through a reduction process is somewhat different, more intense and rich than straight tomatoes.

Indeed, canned tomatoes tend to be surrounded with juice that is more reduced than would occur naturally. Which is part of why they are used against fresh tomatoes.