Sanskrit

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Samskruta, Samskrutha, Sanskrita etc...

It depends upon which transliteration scheme you use. There is nothing correct about any of the above transliterations.

Rishi and Rushi both have the vowel Ri(or Ru) which are rendered incorrectly. It is neither Ri nor Ru but something in between. Remember Maha+Rishi becomes Maharshi after the Sandhi, it doesn't become Maharishi or Maharushi.

About Sam vs San in Sanskrita or Samskrita, the Anuswar or the dot is kept for nasals appearing before the consonants that don't fall in the five major consonant groups. If any consonant falls in the five major groups (ka-vrga, cha-varga ... pa-varga) then any nasal coming before such a consonant takes the form of nasal of that group.

Consonant "sa" is not in the five major groups, it is an "Antahstha". That is why a dot or an "Anuswar" is used in writing Sanskrita. But the nasal is "definitely not" "m" as in Samskrita!! That would work if "s" were labial or "Pa-varg". But "s" is actually a dental consonant also called "dantya sa". The nasal of the dantya group is close to "n" and not "m" which is of labial group. So of all possible spellings Sanskrita is more correct than samskrita.

About "tha" vs "ta" again it depends upon your transliteration scheme. In regular transliteration "t" stands for dental ta-varga, while a t with a dot under it stands for retroflex Ta-vrga. In Itrans scheme the dental "t" is transliterated as "t" while the retroflex t is written as capital "T". In north Indian casual transliteration both dental and retroflex "t" are written simply as "t". Since english doesn't distinguish between these two classes of consonants. In north "tha" is kept for aspirated version of the consonant such as in swastha (healthy). In south "th" is often used for dental "ta" and "t" for retroflex "ta". There is nothing right or wrong about any of these choices. These are the wages of trying to use english to write Sanskrit which is far superior in linguistic phonetic aspects.