A logic analyzer is not going to give you a waveform like the image you posted, with rising/falling edges, but a digitized square wave version of the signal data. Do you need to see the actual waveform?
Does logic analyzer decode the actual waveform? Or does it just convert the waveform to digitized square waves?
Cheap logic analyses won't help you at all. They will try to interpret the signal on the cable as simple logic ones and zeros. Which they aren't. 10 BASE-T uses a differential signal. You would have to convert that to single-ended for a cheap analyser. Further, 10 BASE-T uses a number of specific waveforms to performs some operations, like link tests and signaling idle. What a cheap analyses makes of these? No idea. The actual data is using Manchester code for the line encoding, where they have taken the high-frequency edges off.
The tools commonly available to decode such a signal are called NICs - network interface controllers. Your PC already includes one or more. You probably know it as the network plug on your PC. Plug the cable in, run a software like Wireshark and get more signal decoding and interpretation than you can shake a stick at.