Actually, the saw you've got (14" 11tpi hardpoint) will do just fine for cutting 15mm wood. If a saw has sharp teeth, it'll cut wood. Some saws are better at some cutting jobs than others (see below), but when needs must they can all do cuts they weren't really intended for - they might not be as quick, or as clean-cutting, as the perfect saw, but they'll cut.
The reason a saw is called a 'backsaw' is because it has a strengthening spine along it's back - the fat bit at the top of the sawblade. That's there to stiffen the blade, so that it can be made of thinner steel than the longsaws. The thin blade means more precise cuts when sawing joints such as dovetails and tenons, and makes for quicker sawing because the thinner kerf needs less waste removed than a thicker one would.
Way back in 18th and 19th centuries, when there were few woodworking machines around, the backsaws were developed to enable craftsmen to do their sawing chores quickly and neatly. There were four main types of backsaw. Tenon saws were huge things, about 16" to 20" long, and about 10tpi, designed for cutting the big tenons found in house carpentry and joinery. Sash saws were the same, but smaller; 12" to 14" long and about 12tpi, used for cutting the smaller tenons found in window sashes and cabinet work. Both of these had quite deep blades; about 3" to 4" for sash saws, and up to 5" for tenon saws. (True tenon saws are quite rare these days - the big joints they used to be used for are usually cut by machines these days. Sash saws have become known as tenon saws.) Carcase saws and dovetail saws had shallower blades, about 1 1/2" to 2 1/2", and finer teeth. Carcase saws were 12" to 14" long, about 13 to 15 tpi, and used for cutting dovetails in thicker stock such as cabinet carcase sides, and dovetail saws were 8" to 10" long, 16 to 20 tpi and used for dovetailing thinner stock such as drawer sides. Nowadays, carcase saws are rare, but dovetail saws have survived for small work.
The other thing you may not be aware of is that teeth can be filed to rip (cut with, or along, the grain), or crosscut (cut across the grain). Don't worry too much about this - a rip saw will crosscut, and a crosscut will rip, just not quite as well as the 'proper' filed saw will. Most new saws are crosscut, because a lot of modern work uses man-made boards such as MDF and ply, which cut equally well (or badly!) in whichever direction you saw them.
The one you've got is a good general-purpose all-rounder for all but the fiddliest work. Don't worry about buying a dovetail saw until you need to cut very thin, precise jobs.
To use a backsaw, it helps to hold the work firmly in some way. Clamped in a vice on a bench, or held in a benchhook on a bench is ideal - but not much help if you don't have a bench handy. In that case, clamp the wood if you can. Hold the saw as the sawyer in the picture you posted in your original questio is holding it, with your index finger pointing forwards. If you put all four fingers through the handle hole, you end up with a fist, through which the saw can pivot, and become difficult to steer, so you grip harder to maintain directional control. That tenses the muscles of your forearm, so you're not relaxed, and that makes steerin the saw harder! With the 'finger forward' grip, the index finger lines the saw up, and allows you to relax the forearm muscles and keep much better control. The other three fingers should grip just hard enough to stop the saw falling. Now, grip the work with your off-hand, with the ball of your thumb making a sort of fence to support and guide the sawblade. Line everything up so that the blade is almost parallel to the wood, upright, and your thumb is in contact, keeping it just on the waste side of the line. Start the saw with the toe end of the blade in light contact with the work, and make a gentle forward stroke. It can help to start a saw by 'taking the weight off' - just using your handle hand to almost, but not quite, lift the saw off - imagine you're bringing a plane into land on a runway - gently does it! Once the kerf is established, you can allow the full weight of the saw to do the cutting - don't try to force it or press it down. Just move the saw back and forth with slow, even, full-length strokes, concentrating on following the guide-line with the side of the teeth. Slower strokes means more control - don't try vigourously waggling back and forth like a mad thing, you'll miss your line.
Good luck - any problems, post another question, and we'll sort something out!
