In standard Japanese, 'h' before 'i' or 'y' is pronounced a bit differently (source). In the International Phonetic Alphabet, this sound is written as [ç]. This is the sound you're hearing as kinda like a し. However, it's still distinct from し (which is pronounced as [ɕ]).
The ひ consonant, [ç], does occur in some varieties of English – e.g. British English hue; but, from looking at the Wikipedia article on Hebrew phonology, it looks like you guys don't have it. To understand Japanese, it's important that you get used to hearing the difference between these two sounds. The Google text-to-speech system you posted will do nicely; listen carefully. Notice how ひ sounds softer and closer to English 'h', whereas し sounds noisier and closer to English 'sh'.
Pronouncing this sound in a Japanese way is not vitally important. You can just use English 'he' and 'she' in your accent, and the Japanese will hear it as ひ and し. If you want to perfect your Japanese accent, try to do this: whisper the vowel /i/ forcefully, then change the whisper to the voiced vowel. (While whispering, be sure to have your tongue already in the /i/ position).
By the way, if you want to pronounce the し in a more Japanese manner, try sticking the tip of your tongue behind your lower front teeth.
